diff --git "a/9dE1T4oBgHgl3EQfCQJ2/content/tmp_files/load_file.txt" "b/9dE1T4oBgHgl3EQfCQJ2/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/9dE1T4oBgHgl3EQfCQJ2/content/tmp_files/load_file.txt" @@ -0,0 +1,608 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf,len=607 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='02862v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='MG] 7 Jan 2023 AN INTEGER PARALLELOTOPE WITH SMALL SURFACE AREA ASSAF NAOR AND ODED REGEV ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We prove that for any n ∈ N there is a convex body K ⊆ Rn whose surface area is at most n 1 2 +o(1), yet the translates of K by the integer lattice Zn tile Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' INTRODUCTION Given n ∈ N and a lattice Λ ⊆ Rn, a convex body K ⊆ Rn is called a Λ-parallelotope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [12]) if the translates of K by elements of Λ tile Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', Rn = Λ+K = � x∈Λ(x +K ), and the interior of (x +K )∩(y +K ) is empty for every distinct x, y ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' One calls K a parallelotope (parallelogon if n = 2 and parallelohedron if n = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' some of the literature calls a parallelotope in Rn and n-dimensional parallellohedron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [1, 11]) if it is a Λ-parallelotope for some lattice Λ ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We call a Zn-parallelotope an integer parallelotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The hypercube [− 1 2, 1 2]n is an integer parallelotope whose surface area equals 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By [16, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='2], for every n ∈ N there exists an integer parallelotope K ⊆ Rn whose surface area is smaller than 2n by a universal constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Specifically, the surface area of the integer parallelotope K that was considered in [16] satisfies voln−1(∂K ) ⩽ σ(n +O(n2/3)), where σ = 2�∞ s=1(s/e)s/(s3/2s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=') ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='23721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' To the best of our knowledge, this is the previously best known upper bound on the smallest possible surface area of an integer parallelotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The main result of the present work is the following theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For every n ∈ N there exists an integer parallelotope whose surface area is n 1 2 +o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Because the covolume of Zn is 1, the volume of any integer parallelotope K ⊆ Rn satisfies voln(K ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, by the isoperimetric inequality we have1 voln−1(∂K ) ⩾ voln−1(Sn−1) voln(Bn) n−1 n ≍ � n, (1) where Bn def = {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',xn) ∈ Rn : x2 1 +···+ x2 n ⩽ 1} denotes the Euclidean ball and Sn−1 def = ∂Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Thanks to (1), Theorem 1 is optimal up to the implicit lower order factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' It remains open to determine whether this lower-order factor could be removed altogether, namely to answer the following question: Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For every n ∈ N, does there exist an integer parallelotope K ⊆ Rn with voln−1(∂K ) ≍ �n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Question 2 goes back to [24], though such early investigations were (naturally, from the perspective of crystallography) focused on n = 3 and asked for the exact value of the smallest possible surface area of a parallelohedron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' see Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='5 in [5] and the historical discussion in the paragraph that precedes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The corresponding question about precisely determining the minimum perimeter when n = 2 was answered in [7] (its solution for general parallelogons rather than integer parallelogons is due to [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' see also [22], which treats tiles that need not be convex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Finding the exact minimum when n = 3 remains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' was supported by NSF grant DMS-2054875, BSF grant 201822, and a Simons Investigator award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' was supported by NSF grant CCF-1320188 and a Simons Investigator award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 1We use the following conventions for asymptotic notation, in addition to the usual O(·),o(·),Ω(·),Θ(·) notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For a,b > 0, by writing a ≲ b or b ≳ a we mean that a ⩽ Cb for a universal constant C > 0, and a ≍ b stands for (a ≲ b)∧(b ≲ a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If we need to allow for dependence on parameters, we indicate it by subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For example, in the presence of an auxiliary parameter ε, the notation a ≲ε b means that a ⩽ C(ε)b, where C(ε) > 0 may depend only on ε, and analogously for a ≳ε b and a ≍ε b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 1 open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' we will not review the substantial literature on this topic, referring instead to the monograph [4] (see also [28] for an exact solution of a different isoperimetric-type question for parallelohedra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The higher dimensional asymptotic nature of Question 2 differs from the search for exact minimizers in lower dimensions on which the literature has focused, but it is a natural outgrowth of it and it stands to reason that it was considered by researchers who worked on this topic over the past centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Never- theless, we do not know of a published source that mentions Question 2 prior to the more recent interest in this topic that arose due to its connection to theoretical computer science that was found in [16] and were pursued in [33, 25, 3, 26, 6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' specifically, Question 2 appears in [6, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In [25] it was proved that Question 2 has a positive answer if one drops the requirement that the tiling set is convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', by [25, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='1] for every n ∈ N there is a compact set Ω ⊆ Rn such that Rn = Zn+Ω, the interior of (x + Ω) ∩ (y + Ω) is empty for every distinct x, y ∈ Zn, and voln−1(∂Ω) ≲ �n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' see also the proof of this result that was found in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The lack of convexity of Ω is irrelevant for the applications to computational complexity that were found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The proofs in [25, 3] produce a set Ω that is decidedly non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Our proof of Theorem 1 proceeds via an entirely different route and provides a paralletotope whose surface area comes close to the guarantee of [25] (prior to [25], the best known upper bound on the smallest possible surface area of a compact Zn-tiling set was the aforementioned 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='23721n of [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' While it could be tempting to view the existence of the aforementioned compact set Ω as evidence for the availability of an integer parallelotope with comparable surface area, this is a tenuous hope be- cause the convexity requirement from a parallelotope imposes severe restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In particular, by [30] for every n ∈ N there are only finitely many combinatorial types of parallelotopes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='2 In fact, by com- bining [10, Section 6] with [30, 36] we see that K ⊆ Rn is a parallelotope if and only if K is a centrally symmetric polytope, all of the (n − 1)-dimensional faces of K are centrally symmetric, and the orthog- onal projection of K along any of its (n − 2)-dimensional faces is either a parallelogram or a centrally symmetric hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Of course, Theorem 1 must produce such a constrained polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' To understand how this is achieved, it is first important to stress that this becomes a straightforward task if one only asks for a parallelotope with small surface area rather than for an integer parallelotope with small surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Namely, it follows easily from the literature that for every n ∈ N there exist a rank n lattice Λ ⊆ Rn whose covolume is 1 and a Λ-parallelotope K ⊆ Rn that satisfies voln−1(∂K ) ≲ �n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, by [34] there is a rank n lattice Λ ⊆ Rn of covolume 1 whose packing radius is at least c�n, where c > 0 is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let K be the Voronoi cell of Λ, namely K consists of the points in Rn whose (Euclidean) distance to any point of Λ is not less than their distance to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, K is a Λ-parallelotope, voln(K ) = 1 since the covolume of Λ is 1, and K ⊇ c�nBn since the packing radius of Λ is at least c�n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, the surface area of K is at most c−1�n by the following simple lemma that we will use multiple times in the proof of Theorem 1: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix n ∈ N and R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose that a convex body K ⊆ Rn satisfies K ⊇ RBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, voln−1(∂K ) voln(K ) ⩽ n R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Lemma 3 is known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' for completeness we will present its short proof in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Even though the packing radius of Zn is small, the above observation drives our inductive proof of Theorem 1, which proceeds along the following lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix m ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n−1} and let V be an m-dimensional subspace of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If the lattice V ⊥ ∩Zn has rank n −m and its packing radius is large, then Lemma 3 yields a meaningful upper bound on the (n −m −1)-dimensional volume of the boundary of the Voronoi cell of V ⊥ ∩Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We could then consider the lattice Λ ⊆ V which is the orthogonal projection of Zn onto V , and inductively obtain a Λ-parallelotope (residing within V ) for which the (m −1)-dimensional volume of its boundary is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By considering the product (with respect to the identification of Rn with V ⊥ ×V ) of the two convex bodies thus obtained, we could hope to get the desired integer parallelotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 2Thus, just for the sake concreteness (not important for the present purposes): Since antiquity it was known that there are 2 types of parallelogons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' by [13] there are 5 types of parallelohedra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' by [8, 35] there are 52 types of 4-dimensional parallelotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 2 There are obvious obstructions to this plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The subspace V must be chosen so that the lattice V ⊥∩Zn is sufficiently rich yet it contains no short nonzero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Furthermore, the orthogonal projection Λ of Zn onto V is not Zm, so we must assume a stronger inductive hypothesis and also apply a suitable “correction” to Λ so as to be able to continue the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' It turns out that there is tension between how large the packing radius of V ⊥∩Zn could be, the loss that we incur due to the aforementioned correction, and the total cost of iteratively applying the procedure that we sketched above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Upon balancing these constraints, we will see that the best choice for the dimension m of V is m = n exp(−Θ( � logn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The rest of the ensuing text will present the details of the implementation of this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' PROOF OF THEOREM 1 Below, for each n ∈ N the normed space ℓn 2 = (Rn,∥·∥ℓn 2 ) will denote the standard Euclidean space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', ∀x = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',xn) ∈ Rn, ∥x∥ℓn 2 def = � x2 1 +···+ x2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The standard scalar product of x, y ∈ Rn will be denoted 〈x, y〉 def = x1y1+···+xnyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The coordinate basis of Rn will be denoted e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',en, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', for each i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} the ith entry of ei is 1 and the rest of the coordinates of ei vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We will denote the origin of Rn by 0 = (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For 0 < s ⩽ n, the s-dimensional Hausdorff measure on Rn that is induced by the ℓn 2 metric will be denoted by vols(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In particular, if K ⊆ Rn is a convex body (compact and with nonempty interior), then the following identity holds (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [27]): voln−1(∂K ) = lim δ→0+ voln(K +δBn)−voln(K ) δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (2) If V is a subspace of Rn, then its orthogonal complement (with respect to the ℓn 2 Euclidean structure) will be denoted V ⊥ and the orthogonal projection from Rn onto V will be denoted ProjV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' When treating a subset Ω of V we will slightly abuse notation/terminology by letting ∂Ω be the boundary of Ω within V , and similarly when we will discuss the interior of Ω we will mean its interior within V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' This convention results in suitable interpretations of when K ⊆ V is a convex body or a parallelohedron (with respect to a lattice of V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The variant of (2) for a convex body K ⊆ V becomes voldim(V )−1(∂K ) = lim δ→0+ voldim(V ) � K +δ(V ∩Bn) � −voldim(V )(K ) δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (3) Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Since K ⊇ RBn, for every δ > 0 we have K +δBn ⊆ K + δ R K = � 1+ δ R �� R R +δK + δ R +δK � = � 1+ δ R � K , (4) where the last step of (4) uses the fact that K is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, voln−1(∂K ) (2) = lim δ→0+ voln(K +δBn)−voln(K ) δ (4) ⩽ lim δ→0+ � 1+ δ R �n −1 δ voln(K ) = n R voln(K ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ The sequence {Q(n)}∞ n=1 that we introduce in the following definition will play an important role in the ensuing reasoning: Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For each n ∈ N let Q(n) be the infimum over those Q ⩾ 0 such that for every lattice Λ ⊆ Zn of rank n there exists a Λ-parallelotope K ⊆ Rn that satisfies voln−1(∂K ) voln(K ) ⩽Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (5) As voln(K ) = 1 for any integer parallelotope K ⊆ Rn, Theorem 1 is a special case of the following result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' There exists a universal constant C ⩾ 1 such that Q(n) ≲ �neC� logn for every n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The following key lemma is the inductive step in the ensuing proof of Theorem 5 by induction on n: 3 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix m,n,s ∈ N with s ⩽ m ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose that B ∈ Mm×n(Z) is an m-by-n matrix all of whose entries are integers such that B has rank m and any s of the columns of B are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, Q(n) ⩽ 2(n −m) �s +Q(m)∥B∥ℓn 2 →ℓm 2 , where ∥·∥ℓn 2 →ℓm 2 denotes the operator norm from ℓn 2 to ℓm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The fact that Theorem 5 treats any sublattice of Zn of full rank (recall howQ(n) is defined), even though in Theorem 1 we are interested only in Zn itself, provides a strengthening of the inductive hypothesis that makes it possible for our proof of Lemma 6 to go through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If Λ is an arbitrary full rank sublattice of Zn, then a Λ-parallelotope K ⊆ Rn need no longer satisfy voln(K ) = 1, so the inductive hypothesis must incorporate the value of voln(K ), which is the reason why we consider the quantity voln−1(∂K )/voln(K ) in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Observe that this quantity is not scale-invariant, so it might seem somewhat unnatural to study it, but it is well-suited to the aforementioned induction thanks to the following simple lemma: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix m,n ∈ N and an m-dimensional subspace V of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let O ⊆ V ⊥ be an open subset of V ⊥ and let G ⊆ V be an open subset of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, for Ω = O +G we have voln−1(∂Ω) voln(Ω) = voln−m−1(∂O) voln−m(O) + volm−1(∂G) volm(G) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (6) Furthermore, if T : Rm → V is a linear isomorphism and K ⊆ Rm is a convex body, then volm−1(∂T K ) volm(T K ) ⩽ volm−1(∂K ) volm(K ) ∥T −1∥(V,∥·∥ℓn 2 )→ℓm 2 , (7) where ∥·∥(V,∥·∥ℓn 2 )→ℓm 2 is the operator norm from V , equipped with the norm inherited from ℓn 2 , to ℓm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For (6), note that since O ⊥ G we have voln(Ω) = voln−m(O)volm(G), and ∂Ω = (∂O +G)∪(O +∂G) where voln−1((∂O +G)∩(O +∂G)) = 0, so voln−1(∂Ω) = voln−m−1(∂O)volm(G)+voln−m(O)volm−1(∂G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For (7), denote ρ = ∥T −1∥(V,∥·∥ℓn 2 )→ℓm 2 , so that T −1(V ∩Bn) ⊆ ρBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, ∀δ ∈ R, T K +δ(V ∩Bn) = T � K +δT −1(V ∩Bn) � ⊆ T (K +δρBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By combining this inclusion with (3), we see that volm−1(∂T K ) ⩽ lim δ→0+ volm � T (K +δρBm) � −volm(T K ) δ ⩽ det(T ) lim δ→0+ volm(K +δρBm)−volm(K ) δ (2) = det(T )volm−1(∂K )ρ = volm(T K ) volm(K ) volm−1(∂K )ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We stated Lemma 7 with K being a convex body since that is all that we need herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' However, the proof does not rely on its convexity in an essential way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' all that is needed is that K is a body in Rm whose boundary is sufficiently regular so that the identity (2) holds (with n replaced by m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Any matrix B as in Lemma 6 must have a row with at least n/m nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, otherwise the total number of nonzero entries of B would be less than m(n/m) = n, so at least one of the n columns B would have to vanish, in contradiction to the assumed linear independence (as s ⩾ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Thus, there exists j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m} such that at least ⌈n/m⌉ of the entries of B∗e j ∈ Rn do not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Those entries are integers, so ∥B∗e j∥ℓn 2 ⩾ � ⌈n/m⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, the quantity ∥B∥ℓn 2 →ℓm 2 = ∥B∗∥ℓm 2 →ℓn 2 in (6) cannot be less than � ⌈n/m⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Given m,n ∈ N and C > 1, what is the order of magnitude of the largest s = s(m,n,C) ∈ N for which there exists B ∈ Mm×n(Z) such that any s of the columns of B are linearly independent and ∥B∥ℓn 2 →ℓm 2 ⩽C � n m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The following lemma is a step towards Question 9 that we will use in the implementation of Lemma 6: 4 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose that m,n ∈ N satisfy 4 ⩽ m ⩽ n and n ⩾ (m logm)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' There exist s ∈ N with s ≳ m2/n and B ∈ Mm×n(Z) of rank m such that any s of the columns of B are linearly independent and ∥B∥ℓn 2 →ℓm 2 ≲ � n m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Lemma 10 suffices for our purposes, but it is not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We will actually prove below that in the setting of Lemma 10 for every 0 < ε ⩽ 1 there exist s ∈ N with s ≳ m1+ε/nε = m(m/n)ε ⩾ m2/n and B ∈ Mm×n(Z) of rank m such that any s of the columns of B are linearly independent and ∥B∥ℓn 2 →ℓm 2 ≲ε � n/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' While Question 9 arises naturally from Lemma 6 and it is interesting in its own right, fully answering Question 9 will not lead to removing the o(1) term in Theorem 1 altogether;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' the bottleneck in the ensuing reasoning that precludes obtaining such an answer to Question 2 (if true) is elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof of Theorem 5 assuming Lemma 6 and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We will proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In prepara- tions for the base of the induction, we will first record the following estimate (which is sharp when the lattice is Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The Voronoi cell of a rank n sublattice Λ of Zn, namely the set K = � x ∈ Rn : ∀y ∈ Λ, ∥x∥ℓn 2 ⩽ ∥x − y∥ℓn 2 � , is a Λ-parallelotope that satisfies K ⊇ 1 2Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, if y ∈ Λ∖{0}, then ∥y∥ℓn 2 ⩾ 1 since y ∈ Zn∖{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, ∀x ∈ 1 2Bn, ∥x − y∥ℓn 2 ⩾ ∥y∥ℓn 2 −∥x∥ℓn 2 ⩾ ∥x∥ℓn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By Lemma 3, it follows that voln−1(∂K )/voln(K ) ⩽ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' This gives the (weak) a priori bound Q(n) ⩽ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix n ∈ N and suppose that there exists m ∈ N satisfying 4 ⩽ m ⩽ n and n ⩾ (m logm)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By using Lemma 6 with the matrix B from Lemma 10 we see that there is a universal constant κ ⩾ 4 for which Q(n) ⩽ κ � n 3 2 m +Q(m) � n m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (8) We will prove by induction on n ∈ N the following upper bound on Q(n), thus proving Theorem 5: Q(n) ⩽ 4κ � ne � 2(logn)log(2κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (9) If n ⩽ 4κ2, then by the above discussion Q(n) ⩽ 2n ⩽ 4κ�n, so that (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If n > 4κ2, then define m def = � ne−� 2(logn)log(2κ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (10) It is straightforward to verify that this choice of m satisfies 4 ⩽ m < n and n ⩾ (m logm)/4 (with room to spare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Therefore (8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Using the induction hypothesis, it follows that Q(m) � n m ⩽ 4κ � ne � 2(logm)log(2κ) (10) ⩽ 4κ � ne � 2 � logn−� 2(logn)log(2κ) � log(2κ) ⩽ 4κ � ne �� 2logn−� log(2κ) �� log(2κ) = 2 � ne � 2(logn)log(2κ), (11) where the penultimate step of (11) uses the inequality � a −b ⩽ �a − b/(2�a), which holds for every a,b ∈ R with a ⩾ b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' in our setting a = logn and b = � 2(logn)log(2κ) and a > b because we are now treating the case n > 4κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' A substitution of (11) into (8), while using that m ⩾ 1 2n exp � − � 2(logn)log(2κ) � holds thanks to (10), gives (9), thus completing the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ We will next prove Lemma 6, which is the key recursive step that underlies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We will start with the following two elementary observations to facilitate the ensuing proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Denote the span of the rows of B by V = B∗Rm ⊆ Rn and notice that dim(V ) = m as B is assumed to have rank m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose that Λ is a lattice of rank n that is contained in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Firstly, we claim that the rank of the lattice V ⊥ ∩Λ equals n −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, we can write V ⊥ ∩Λ = C(Zn ∩C−1V ⊥) where C is an invertible matrix with integer entries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', C ∈ Mn(Z) ∩GLn(Q), such that Λ = CZn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Furthermore, V ⊥ = Ker(B), so 5 the dimension over Q of Qn ∩V ⊥ equals n − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' As C−1 ∈ GLn(Q), it follows that C−1V ⊥ contains n − m linearly independent elements of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Secondly, we claim that the orthogonal projection ProjV Λ of Λ onto V is a discrete subset of V , and hence is a lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' its rank will then be dim(V ) = m because we are assuming that span(Λ) = Rn, so span(ProjV Λ) = ProjV (span(Λ)) = ProjV (Rn) = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We need to check that for any {x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='} ⊆ Λ such that limi→∞ProjV xi = 0 there is i0 ∈ N such that ProjV xi = 0 whenever i ∈ {i0,i0 +1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, as V ⊥ = Ker(B) we have Bx = BProjV x for every x ∈ Rn, so limi→∞Bxi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' But, Bxi ∈ Zm for every i ∈ N because B ∈ Mm×n(Z) and xi ∈ Λ ⊆ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, there is i0 ∈ N such that Bxi = 0 for every i ∈ {i0,i0 +1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', xi ∈ Ker(B) = V ⊥ and hence ProjV xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let K1 ⊆ V ⊥ be the Voronoi cell of V ⊥ ∩Λ, namely K1 = {x ∈ V ⊥ : ∀y ∈ V ⊥ ∩Λ, ∥x∥ℓn 2 ⩽ ∥x − y∥ℓn 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If y = (y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', yn) ∈ V ⊥ = Ker(B), then y1Be1 +··· + ynBen = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By the assumption on B, this implies that if also y ̸= 0, then |{i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} : yi ̸= 0}| > s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, as the entries of elements of Λ are integers, ∀y ∈ (V ⊥ ∩Λ)∖{0}, ∥y∥ℓn 2 > � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, if x ∈ �s 2 (V ⊥ ∩Bn), then ∀y ∈ (V ⊥ ∩Λ)∖{0}, ∥x − y∥ℓn 2 ⩾ ∥y∥ℓn 2 −∥x∥ℓn 2 > � s − �s 2 = �s 2 ⩾ ∥x∥ℓn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' This means that K1 ⊇ �s 2 (V ⊥ ∩Bn), and therefore by Lemma 3 we have voln−m−1(∂K1) voln−m(K1) ⩽ n −m 1 2 �s = 2(n −m) �s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (12) Next, fix i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By the definition of V , the i’th row B∗ei of B belongs to V , so ∀(x,i) ∈ Rn ×{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}, 〈x,B∗ei〉 = 〈ProjV x,B∗ei〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (13) Since all of the entries of B are integers, it follows that ∀(x,i) ∈ Zn ×{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}, 〈BProjV x,ei〉 = 〈ProjV x,B∗ei〉 (13) = 〈x,B∗ei〉 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In other words, BProjV Zn ⊆ Zm, and hence the lattice BProjV Λ is a subset of Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Furthermore, B is injective on V because Ker(B) = V ⊥, so BProjV Zn is a rank m sublattice of Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By the definition of Q(m), it follows that there exists a BProjV Λ-parallelotope K 0 2 ⊆ Rm such that volm−1(∂K 0 2 ) volm(K 0 2 ) ⩽ Q(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (14) Because V ⊥ = Ker(B) and the rank of B is m = dim(V ), the restriction B|V of B to V is an isomorphism between V and Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Letting T : Rm → V denote the inverse of B|V , define K2 = T K 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By combining (the second part of) Lemma 7 with (14), we see that volm−1(∂K2) volm(K2) ⩽Q(m)∥B∥ℓn 2 →ℓm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (15) Let K = K1 +K2 ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By combining (the first part of) Lemma 7 with (12) and (15), we have voln−1(∂K ) voln(K ) ⩽ 2(n −m) �s +Q(m)∥B∥ℓn 2 →ℓm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, the proof of Lemma 6 will be complete if we check that K is a Λ-parallelotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Our construction ensures by design that this is so, as K1 is a (V ⊥ ∩Λ)-parallelotope and K2 is a ProjV Λ-parallelotope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' veri- fying this fact is merely an unravelling of the definitions, which we will next perform for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix z ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' As Rm = BProjV Λ+K 0 2, there is x ∈ Λ with BProjV z ∈ BProjV x+K 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Apply T to this inclusion and use that TB|V is the identity mapping to get ProjV z ∈ ProjV x +K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Next, V ⊥ = K1 +V ⊥ ∩Λ since K1 is the Voronoi cell of V ⊥ ∩Λ, so there is y ∈ V ⊥ ∩Λ such that ProjV ⊥z −ProjV ⊥x ∈ y +K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, z = ProjV ⊥z +ProjV z ∈ ProjV ⊥x + y +K1 +ProjV x +K2 = x + y +K ∈ Λ+K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, Λ+K = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 6 It remains to check that for every w ∈ Λ∖{0} the interior of K does not intersect w +K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, by the definition of K , if k belongs to the interior of K , then k = k1+k2, where k1 belongs to the interior of K1 and k2 belongs to the interior of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Since B is injective on K2 ⊆ V , it follows that Bk2 belongs to the interior of BK2 = K 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If ProjV w ̸= 0, then BProjV w ∈ BProjV Λ∖ {0}, so because K 0 2 is a BProjV Λ-parallelotope, Bk2 ∉ BProjV w + K 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By applying T to is inclusion, we see that k2 ∉ ProjV w + K2, which implies that k ∉ w +K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' On the other hand, if ProjV w = 0, then w ∈ (V ⊥ ∩Λ)∖{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Since K1 is a V ⊥ ∩Λ-parallelotope, it follows that k1 ∉ w +K1, so k ∉ w +K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ To complete the proof of Theorem 5, it remains to prove Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For ease of later reference, we first record the following straightforward linear-algebraic fact: Observation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix m,n,s ∈ N with s ⩽ m ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose that there exists A ∈ Mm×n(Z) such that any s of the columns of A are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, there also exists B ∈ Mm×n(Z) such that any s of the columns of B are linearly independent, B has rank m, and ∥B∥ℓn 2 →ℓm 2 ⩽ � 1+∥A∥2 ℓn 2 →ℓm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let r ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m} be the rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By permuting the rows of A, we may assume that its first r rows, namely A∗e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',A∗er ∈ Rn are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Also, since we can complete A∗e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',A∗er to a basis of Rn by adding n−r vectors from {e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',en} ⊆ Rn, by permuting the columns of A, we may assume that the vectors A∗e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',A∗er,er+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',em ∈ Rn are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let B ∈ Mm×n(Z) be the matrix whose rows are A∗e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',A∗er,er+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',em, so that B has rank m by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Also, ∀x ∈ Rn, ∥Bx∥2 ℓm 2 = r� i=1 (Ax)2 i + m � j=r+1 x2 j ⩽ � ∥A∥2 ℓn 2 →ℓm 2 +1 � ∥x∥2 ℓn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Therefore (16) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' It remains to check that any s of the columns of B are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, fix S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} with |S| = s and {αj }j∈S ⊆ R such that � j∈S αjBi j = 0 for every i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' In particular, � j∈S αjAi j = 0 for every i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If k ∈ {r +1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}, then since the k’th row of A is in the span of the first r rows of A, there exist βk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',βkr ∈ R such that Ak j = �r i=1βkiAi j for every j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Conse- quently, � j∈S αjAk j = �r i=1βki � j∈S αjAi j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' This shows that � j∈S αjAi j = 0 for every i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By the assumed property of A, this implies that αj = 0 for every j ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ The following lemma is the main existential statement that underlies our justification of Lemma 10: Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' There exists a universal constant c > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let d,m,n ⩾ 3 be integers that satisfy d ⩽ m ⩽ n and n ⩾ (m logm)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Suppose also that s ∈ N satisfies s ⩽ c d � md n2 � 1 d−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (17) Then, there exists an m-by-n matrix A ∈ Mm×n({0,1}) with the following properties: Any s of the columns of A are linearly independent over the field Z/(2Z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Every column of A has at most d nonzero entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Every row of A has at most 5dn/m nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The ensuing proof of Lemma 12 consists of probabilistic reasoning that is common in the literature on Low Density Parity Check (LDPC) codes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' it essentially follows the seminal work [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' While similar considerations appeared in many places, we could not locate a reference that states Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='3 A pecu- liarity of the present work is that, for the reason that we have seen in the above deduction of Theorem 5 from Lemma 6 and Lemma 10, we need to choose a nonstandard dependence of m on n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' recall (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 3The standard range of parameters that is discussed in the LDPC literature is, using the notation of Lemma 12, either when m ≍ n, or when s,d are fixed and the pertinent question becomes how large n can be as m → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' sharp bounds in the former case are due to [18] and sharp bounds in the latter case are due to [29, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Investigations of these issues when the parameters have intermediate asymptotic behaviors appear in [15, 14, 2, 9, 21, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 7 In the course of the proof of Lemma 12 we will use the following probabilistic estimate: Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let {W (t) = (W (t,1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',W (t,m))}∞ t=0 be the standard random walk on the discrete hypercube {0,1}m, starting at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Thus, W (0) = 0 and for each t ∈ N the random vector W (t) is obtained from the random vector W (t −1) by choosing an index i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m} uniformly at random and setting W (t) = � W (t −1,1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',W (t −1,i −1),1−W (t −1,i),W (t −1,i +1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',W (t −1,m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, Prob[W (t) = 0] ⩽ 2(t/m)t/2 for every t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If t is odd, then Prob[W (t) = 0] = 0, so suppose from now that t is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Let P ∈ M{0,1}m×{0,1}m(R) denote the transition matrix of the random walk W , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', ∀f : {0,1}m → R, ∀x ∈ {0,1}m, Pf (x) = 1 m m � i=1 f (x +ei mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Then, Prob[W (t) = 0] = (Pt)00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By symmetry, all of the 2m diagonal entries of Pt are equal to each other, so (Pt)00 = Trace(Pt)/2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For every S ⊆ {0,1}m, the Walsh function (x ∈ {0,1}m) �→ (−1) � i∈S xi is an eigen- vector of P whose eigenvalue equals 1−2|S|/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, Prob[W (t) = 0] = 1 2m Trace(Pt) = 1 2m m � k=0 � m k �� 1− 2k m �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (18) Suppose that β1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',βm are independent {0,1}-valued unbiased Bernoulli random variables, namely, Prob[βi = 0] = Prob[βi = 1] = 1/2 for any i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By Hoeffding’s inequality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [37, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='6]), ∀u ⩾ 0, Prob ����� m � i=1 � βi − 1 2 ����� ⩾ u � ⩽ 2e− 2u2 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (19) Observing that the right hand side of (18) is equal to the expectation of � 1− 2 m �m i=1βi �t, we see that Prob[W (t) = 0] (18) = � − 2 m �t E �� m � i=1 � βi − 1 2 ��t� = � 2 m �t �∞ 0 tut−1Prob ����� m � i=1 � βi − 1 2 ����� ⩾ u � du (19) ⩽ 2t � 2 m �t �∞ 0 ut−1e− 2u2 m du = 2 � 2 m � t 2 � t 2 � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' ⩽ 2 � 2 m � t 2 � t 2 � t 2 = 2 � t m � t 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ With Lemma 13 at hand, we can now prove Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consider the random matrix A ∈ Mm×n({0,1}) whose columns are independent iden- tically distributed copies W1(d),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',Wn(d) of W (d), where W (0) = 0,W (1),W (2),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' is the standard ran- dom walk on {0,1}m as in Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By design, this means that each column of A has at most d nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fixing (i, j) ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m}×{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n}, if Wj(d,i) = 1, then in at least one of the d steps of the random walk that generated Wj(d) the ith coordinate was changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The probability of the latter event equals 1−(1−1/m)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, Prob[Wj(d,i) = 1] ⩽ 1−(1−1/m)d ⩽ d/m and therefore for every fixed S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n}, the probability that Wj(d,i) = 1 for every j ∈ S is at most (d/m)|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Consequently, the probability that all of the rows of A have at most ℓ = ⌈4dn/m⌉ nonzero entries is at least 1−m � n ℓ �� d m �ℓ ⩾ 1−m �en ℓ �ℓ � d m �ℓ = 1−m �edn mℓ �ℓ ⩾ 1−m �e 4 �4logm ⩾ 1 3, where the first step is an application of Stirling’s formula, the penultimate step uses ℓ ⩾ 4dn/m and the assumption n ⩾ (m logm)/d, and the final step holds because m ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' It therefore suffices to prove that with probability greater than 2/3 the vectors {Wi (d)}i∈S ⊆ {0,1}m are linearly independent over Z/(2Z) for every ∅ ̸= S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} with |S| ⩽ s, where s ∈ N satisfies (17) and the universal constant c > 0 that appears in (17) will be specified later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' see (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' So, it suffices to prove that with probability greater than 2/3 we have � i∈S Wi(d) ̸≡ 0 mod 2 for every ∅ ̸= S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} with |S| ⩽ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' 8 Hence, letting D denote the number of ∅ ̸= S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} with |S| ⩽ s that satisfy � i∈S Wi(d) ≡ 0 mod 2, it suffices to prove that 2/3 < Prob[D = 0] = 1−Prob[D ⩾ 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Using Markov’s inequality, it follows that the proof of Lemma 12 will be complete if we demonstrate that E[D] < 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' The expectation of D can be computed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, E[D] = E � � S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} 1⩽|S|⩽s 1{ � i∈S Wi(d)≡0 mod 2} � = s� r=1 � n r � Prob[W (dr) = 0], (20) where we used the fact that � i∈S Wi(d) mod 2 ∈ {0,1}m has the same distribution as W (d|S|) for every ∅ ̸= S ⊆ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' By substituting the conclusion of Lemma 13 into (20) we see that E[D] ⩽ 2 s� r=1 � n r ��dr m � dr 2 ⩽ 2 s� r=1 �ed d 2 r d 2 −1n m d 2 �r , (21) where in the last step we bounded the binomial coefficient using Stirling’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' For every r ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',s}, ed d 2 r d 2 −1n m d 2 ⩽ ed d 2 s d 2 −1n m d 2 (17) ⩽ edc d 2 −1 < 1 7, (22) provided that c < inf d⩾3 � 1 7ed � 2 d−2 ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' (23) Therefore, when (23) holds we may substitute (22) into (21) to get that E[D] < 2�∞ r=1 1 7r = 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ We can now prove Lemma 10, thus concluding the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' We will prove the following stronger statement (Lemma 10 is its special case ε = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If 0 < ε ⩽ 2 and m,n ∈ N satisfy 2 + ⌊2/ε⌋ ⩽ m ⩽ n and n ⩾ (m logm)/(2 + ⌊2/ε⌋), then there exist s ∈ N with s ≳ εm1+ε/nε, and B ∈ Mm×n(Z) such that any s of the columns of B are linearly independent, the rows of B are linearly independent, and ∥B∥ℓn 2 →ℓm 2 ≲ 1 ε � n m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Indeed, apply Lemma 12 with d = 2 + ⌊2/ε⌋ ⩾ 3 (equivalently, d ⩾ 3 is the largest integer such that 2/(d −2) ⩾ ε) to deduce that there exist an integer s with s ≍ 1 d � md n2 � 1 d−2 = m d �m n � 2 d−2 ≍ εm �m n �ε = εm1+ε nε , and a matrix A ∈ Mm×n({0,1}) ⊆ Mm×n(Z) such that any s of the columns of A are linearly independent over Z/(2Z), every column of A has at most d nonzero entries, and every row of A has at most 5dn/m nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' If a set of vectors v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',vs ∈ {0,1}m is linearly independent over Z/(2Z), then it is also linearly independent over R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', letting V ∈ Mm×s({0,1}) denote the matrix whose columns are v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',vs, the latter requirement is equivalent to the determinant of V∗V ∈ Ms({0,1}) being an odd integer, so in particular it does not vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Hence, any s of the columns of A are linearly independent over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Also, ∥A∥ℓn 2 →ℓm 2 ⩽ � max i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',m} n� j=1 |Ai j| � 1 2 � max j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=',n} m � i=1 |Ai j| � 1 2 ⩽ � 5dn m · � d ≍ 1 ε � n m , where the first step is a standard bound which holds for any m-by-n real matrix (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' [20, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Thus, A has all of the properties that we require from the matrix B in Lemma 10, except that we do not know that A has rank m, but Observation 11 remedies this (minor) issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' □ We end by asking the following question: 9 Question 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Fix n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Does there exist an integer parallelotope K ⊆ Rn such that the (n−1)-dimensional area of the orthogonal projection Projθ⊥K of K along any direction θ ∈ Sn−1 is at most no(1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' An application of Cauchy’s surface area formula (see [27, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='5]), as noted in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=', [31, Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content='6], shows that a positive answer to Question 14 would imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Correspondingly, a posi- tive answer to Question 14 with no(1) replaced by O(1) would imply a positive answer to Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' Apart from the intrinsic geometric interest of Question 14, if it had a positive answer, then we would deduce using [31] that there exists an integer parallelotope K ⊆ Rn such that the normed space X whose unit ball is K has certain desirable nonlinear properties, namely, we would obtain an improved random- ized clustering of X and an improved extension theorem for Lipschitz functions on subsets of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE1T4oBgHgl3EQfCQJ2/content/2301.02862v1.pdf'} +page_content=' we refer to [31] for the relevant formulations since including them here would result in 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