Questions tagged [integer-lattices]

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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Can every lattice obtained by orthogonal projection of the cubic lattice?

Assume that $\Lambda\subset\mathbb{Z}^n$ is a full-dimensional lattice with the generator matrix $G\in\mathbb{Z}^{n\times n}$. Can I always embed $\mathbb{R}^n$ into a bigger dimensional vector space, ...
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Proving that $a_i,b_i \to 0$ where $a_{i+1} := |b_i - a_i|$ and $b_{i+1} := |a_i - a_{i+1}|$?

Let $a_0 := 1$ and $b_0 := \sqrt{2}$, and define \begin{align*} a_{i+1} &:= |b_i - a_i| \\ b_{i+1} &:= |a_i - a_{i+1}| \end{align*} Prove that $\lim_{i \to \infty}{a_i} = 0$ and $\lim_{i \to \...
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Natural Density For Subsets of $\mathbb{N}^2$?

Suppose that we have an set $S\subset\mathbb{N}^2$. Taking the natural order on $\mathbb{N}^2$ induced by $f((n,m))=$ max$(\{n,m\})$, what can we say about the "natural density" of S in $\...
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Number of lattice points $7$ units away

A lattice point is a point in space which has all coordinates as integers. How many $3D$ lattice points are exactly $7$ units from the origin? I'm thinking the answer to this is $3$, since we have ...
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Real analytic elliptic functions

First off, I'm sorry that the question is somewhat open-ended. I am interested in functions such as $$f(z;\tau)=\sum_{m,n\in\mathbb{Z}}\frac{1}{(|z+m\tau+n|^2+a^2)^{\frac{3}{2}}}\,,\quad g(z;\tau)=\...
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If $\omega_1/\omega_2$ is not real then there exists a constant $k >0$ such that for all $n,m \in \mathbb Z$, $|m\omega_1+n\omega_2| \geq k(|m|+|n|)$

Let $\omega_1, \omega_2$ be two nonzero complex numbers. If $\omega_1/\omega_2$ is not real, prove that there exists a constant $k >0$ such that for all $n,m \in \mathbb Z$ $|m\omega_1+n\omega_2| \...
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Combinatorial isoperimetric inequality

I am interested in isoperimetric equalities in lattices, more precisely what could be said for surrounding territory in the game of go. Here are some statements or heuristics: To surround territory ...
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Check if two bases form the same lattice

Hi I am currently working on a problem regarding different bases and their respective lattices. I am a physicist, so maybe the question might be trivial (I don't know), but I was not able to find ...
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Voronoi relevant vectors VS shortest vector

Let $L$ be an $n$-dimensional lattice with its Voronoi cell $\mathcal{V}$ is then defined as the set $\{x\in \mathbb R^n:|x|\le |x-v|, \mbox{ for all }v\in L\}$. A vector $v$ is called a Voronoi ...
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Finding an isomorphism between an infinite tree and a subgraph of $\mathbb{Z}^3$

I was wondering if there exists a construction of an infinite tree, with some properties, that is isomorphic to subgraph of $\mathbb{Z}^3$. Notation Let $\Gamma_n$ denote the tree's vertices at ...
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In which dimensions do there exist inequivalent lattices with the same theta function?

Equivalently, "Is a lattice determined by the distances (with multiplicities) of its points from the origin?" By a lattice $L$ I mean a discrete additive subgroup of Euclidean space $\mathbb{...
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Enumerating the square equivalent classes

Fill the $n\times n$ lattice square with natural numbers $\{1,2,\cdots,n\}$ each of which is used exactly $n$ times. Two configurations are in an equivalence class if and only if one is transformed ...
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Second group cohomology $H^2(\mathbb Z^n, \mathbb C^*)=?$

Regard $\mathbb Z^n$ as an abelian group, and let $\mathbb C^* = \mathbb C-\{0\}$. Question: What is the group cohomology $H^2(\mathbb Z^n, \mathbb C^*)$? More specific question is as follows. I have ...
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Problem from Red Book

The following problem is from The Red Book (list of practice problems for undergraduate mathematics competition). Let $k$ denote a positive integer. Determine the number $N(k)$ of triples $(x,y,z)$ of ...
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Minimum number of straight line segments in a polygonal chain covering a square lattice

Let $a:={1,2,\cdots,n}$. Consider $A:=a\times a$ square lattice points on $\mathbf Z^2$. We are to draw a polygonal chain consisting of $m$ straight line segments traversing all points in $A$. What is ...
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Show that there exist lattices $ \Lambda \subseteq \mathbb{R} $ of arbitrarily large dimension.

I'm reading Fundamental Problems of Algorithmic Algebra by Chee-Keng Yap and could not solve the following problem. Exercise 1.1: Show that there exist lattices $ \Lambda \subseteq \mathbb{R} $ of ...
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A shortest k-sequence need not form a basis of the lattice

I'm reading Fundamental Problems of Algorithmic Algebra by Chee-Keng Yap and could not solve the following problem. The shortest k-sequence (where k > 2 is the dimension of the lattice) need not ...
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number of paths between opposite boundaries of a cube

There is a calculation of the number of surface paths (with no backtracking allowed) between opposite corners of a Rubik's cube. I am interested in paths on an $L\times L\times L$ cube, where $L$ is ...
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Grow a shape in the Euclidean plane (or space) by adding unit boxes to the surface at random places. Do we get an almost-ball almost-certainly?

I hope my title asks the question reasonably well, but I'll add some rigor here. Consider a process as follows: at step $ n = 0 $ we start with a set consisting of one point, $ S_0 = \{(0,0)\} $. ...
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Where do the "classical lattices" come from?

I am trying to understand sphere packing problems, and it seems well-known that lattices can be only of certain kinds: the classical types (e.g. $A_n, B_n$, etc.) or the exceptional types (e.g. $E_8$)....
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Legendrian in the cosphere bundle

Let $M$ and $N$ be dual lattices of rank $n$, i.e. $N\cong \text{Hom}(M, \Bbb{Z})$. Let $M_\Bbb{R}=M\otimes \Bbb{R}$ and similarly $N_\Bbb{R}$. Then $T^n\cong M_\Bbb{R}/M$ is the $n$-torus and we can ...
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how can you tell if two lattices are the same

I'm looking at three different definitions of the $E_8$ lattice : $G_a$ is the coding theory version $G_b$ is the root lattice version $G_c$ is the wiki version According to this old question the way ...
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how to go from classical binary code to lattice

I think the (type A) construction is straight forward but I can't find a definitive reference. I'm interested in getting an explicit generator matrix for the lattice $G_L$. The starting point is the ...
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Proof that any element of a free abelian group can be extended to a basis

Let $A$ be a free abelian group of rank $n$, and let $\alpha_1 \in A\setminus \{0\}$ such that $\alpha_1 \not \in kA$ for all $k > 1$. Do there always exist $\alpha_2,\ldots, \alpha_n \in A$ such ...
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Question about algebraic integers and roots of unit

Let $y$ be an algebraic integer in a finite field extension $K:Q$. Hence y is an element in the maximal Z-order $O_K$. The question is if $y$ and all its conjugates have absolute value 1, then y in a ...
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Lattice paths of on $\mathbb{Z^d}$

Consider an infinite lattice on $\mathbb{Z}^d$. Neighbours of a point on the lattice are given as von-Neumann neighbours, i.e a neighbour of a point $(x, y)$ are the set $$ \{(x+1,\ y),(x-1,\ y),(x,\ ...
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Poisson summation formula for lattices not necessarily of full rank

Let $\Lambda$ be an integral lattice (i.e. $\Lambda \subset \mathbb{Z}^n$) in $\mathbb{R}^n$. If $\Lambda$ is of full rank, i.e. the rank of $\Lambda$ is $n$, and $\phi:\mathbb{R}^n \to \mathbb{R}$ is ...
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Path Lengths and Random Walk - Generalization of the Uncut Spaghetti Game

The following is a kind of generalization of the interesting Uncut Spaghetti game (https://mathpickle.com/project/uncut-spaghetti-number-patterns/): Consider a 2D integer lattice where each point has ...
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What is the degeneracy of the "Coxeter plane" for the E8 lattice?

The 240 minimal vectors (roots) of the E8 lattice, projected onto "the" Coxeter plane, are shown here: https://en.wikipedia.org/wiki/E8_(mathematics)#/media/File:E8Petrie.svg and discussed ...
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Lattice embeddings in $\mathbb{Z}^n$

I have a certain class of lattices, and given $\Lambda$ in this class I would like to find obstructions to the existence of a finite-index lattice embedding $\Lambda \rightarrow \mathbb{Z}^n$ for some ...
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5 votes
2 answers
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The number of closed paths in the square lattice $\mathbb{Z}^2$ with length $n$ and starting and ending points at $(0,0)$.

I'm thinking about this problem right now. Problem:Consider a lattice point consisting of $\mathbb{Z}^2$ points. If $n$ is even, i.e., $n=2p$, then Show that the number of closed paths in the square ...
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How many orthogonal pairs of shortest vectors in the Leech lattice?

The Leech Lattice in $\mathbb{R}^{24}$ has 196580 shortest vectors. Let $V$ be the set of all such vectors. For any pair of vectors $x,y \in V$, the inner product $<x,y>$ is an integer between $-...
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Why is the weight lattice a lattice?

For $\mathfrak g$ a complex semisimple Lie algebra, $\mathfrak h \subset \mathfrak g$ a CSA and $R$ the corresponding set of roots we call the following set $\Lambda_w=\{\lambda \in \mathfrak h^*|\...
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Semi-lattices whose Hasse Diagrams are trees after transitive reduction?

Is there a name or anything else known for a semi-lattice whose Hasse Diagram becomes a tree after applying transitive reduction? Trying to find more about it since it comes up in an optimization ...
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3 votes
2 answers
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Is there a rational point in a given open set such that the distance from given rational points to it are all irrational numbers on $\mathbb{R}^2$?

On $\mathbb{R}^2$ there is a nonempty open set $A$ and $n$ rational points $a_1,\ldots,a_n$. Is there a rational point $a$ in $A$ such that $\forall i\in\{1,\ldots,n\}, |a−a_i|$ is an irrational ...
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Is there a rational point in a given open set such that the distancse from given rational points to it are all rational numbers on $R^2$?

On $R^2$ there is a nonempty open set $A$ and n rational points $a_1,...,a_n$. Is there a rational point $a$ in $A$ such that $\forall n\in\{1,..,n\},~|a-a_i|$ is a rational number. My idea is to find ...
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2 votes
1 answer
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An isomorphism of unimodular lattices in $\mathbb{R}^n$

In the proof that $SL_{n}(\mathbb{Z})$ is a lattice in $SL_{n}(\mathbb{R})$, the following isomorphism is used $$SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z}) \cong \{\text{unimodular lattices in } \mathbb{R}^...
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Distance between the ellipsoid and the integer lattice

Let $r_1, r_2, \dots, r_n > 0$ be positive real numbers and let $$ E: \Big(\frac{x_1}{r_1}\Big)^2+\Big(\frac{x_2}{r_2}\Big)^2+\dots+\Big(\frac{x_n}{r_n}\Big)^2 = 1 $$ be the corresponding ellipsoid ...
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In a distributive lattice, which are the equivalence classes of the projectivity relation on prime intervals?

Let $L$ be a lattice (we can assume that it is distributive), according to Birkhoff (page 72): Two intervals of a lattice are called trasposes when they can be written as $[a \wedge b, a]$ and $[b, a \...
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Commutativity of T(n) and R(n) as functions on lattices (Lang Introduction to Modular Forms)

I am currently reading through Lang's Introduction to Modular Forms. In chapter II, he introduces the Hecke Operator as follows. Let $\mathcal{L}$ be the free abelian group generated by the lattices ...
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Which are the latticial properties of the quartered aztec diamond?

The aztec diamond is an area of a 2-dimensional square lattice. The quarter aztec diamond it's a part of this area, it can be seen in the following picture: Triangular arrangement of a 2-dimensional ...
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Projection of a lattice need not be a lattice

I have the following problem: Let $L$ be a $\mathbb{R}^n$ lattice (that is a discrete closed $\mathbb{R}^n$ subgroup). Let $E$ be a vector subspace of $\mathbb{R}^n$ and consider $\pi$ to be ...
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is the intersection with a Lattice still a Lattice?

Given a lattice A in $R^n$, and a subspace B of $R^n$. is the intersection $A \cap B$ a lattice? Thanks
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Unimodular quadratic forms where all values are multiples of $m$

I am interested in unimodular integral positive-definite quadratic forms taking values in multiples of $m$, for some integer $m \geq 2$, which are maps $Q : \mathbb Z^n \to m \mathbb Z_{\geq 0}$. (...
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2 votes
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How do you solve the system $Ax=b$ where $\|x\|_1 \leq \delta$ and $x \in \lbrace 0,1 \rbrace^n$?

During my studies I recently met the problem of finding a binary variable $x \in \lbrace 0,1 \rbrace ^n$ that solves $$Ax=b, \qquad \|x\|_1 \leq \delta$$ I am curious how this system can be solved. ...
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From Automorphism of code to automorphism of lattice

From a code, a lattice can be constructed using many methods. For codes over $\mathbb{F}_2$, there is the straight construction \begin{equation} \Lambda(C) := \{v/\sqrt{2} \ | \ v \in \mathbb{Z}^n ...
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A question about the least common multiple of some numbers.

Let $a_1,\cdots,a_n\in\mathbb N_+$ be positive integers. Let $A:=\mathrm{lcm}(a_1,\cdots,a_n)$ be their least common multiple. Let $d\in\mathbb N_+$ be a positive integer. Define a set $$S_d:=\big\{(...
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Why is the covolume of a lattice is unique?

Given a basis $\mathfrak{B}=\{b_1, b_2, \cdots, b_n\}$ of $\mathbb{R}^n,$ the lattice generated by $\mathfrak{B}$ is the set of all linear combinations with integer coefficients: $$m_1b_1+m_2b_2+\...
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Lattices and fundamental domain

I would like a direction on how to approach this question, without using the lattice generator matrix. Let L \subset \mathbb{Z}^{n} be a finite index subgroup. Show that the volume of a fundamental ...
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If two lattices have the same kissing number and center density, then are they similar?

Two lattices $\Lambda$ and $\Omega$ (the $\mathbb{Z}$-span of a linearly independent set $B\subset \mathbb{R}^n$) are said to be similar if there exist a real orthogonal $n \times n$ matrix $A$ and a ...
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