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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Infinite series over lattice of Gaussian integers

I'm trying to show that the following sum converges to $0$ over the lattice $L = \mathbb{Z}[i]$ of Gaussian integers: $$ 140\times\sum_{\substack{l \in L \\ l\neq 0}} l^{-6} = 0. $$ I don't really ...
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Let $x,y>1$ be coprime integers and $g>0$ a real number such that $g^x,g^y$ are both integers. Is it true that $g\in\mathbb N$?

Let: $x, y\ $ be coprime integers greater than $1$ $g \in \mathbb{R}^+$ $g_,^x \ g^y \in \mathbb{N}$ Proposition: $g \in \mathbb{N}$ I have not managed to prove it. Via the fundamental theorem of ...
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Dimension of an affine space

Let $v=(v_1,\cdots,v_n)\in\mathbb R^n$ and $\Lambda\subset \mathbb R^n$ ($\Lambda$ is a discrete set) with $v\in\Lambda$. Consider the subset of $\mathbb R^n$ given by the set of those $x\in \mathbb R^...
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Relevant Voronoi vector and Shortest Vector

Possibly this is related to the question I asked yesterday here, but I am not very sure about that. Let $L$ be an $n$-dimensional lattice. The Voronoi cell $V(u)$ is then defined as the set $\{x\in \...
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Test if a point on a hexagonal lattice falls on a specified superlattice?

Based on previous answers (1, 2, 3) integers $i, j$ produce a hexagonal lattice using $$x = i + j/2$$ $$y = j \sqrt{3} / 2.$$ From a point $k, l$ I can make a superlattice from integers $I, J$ ...
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Definition of Voronoi relevant vectors

The following definition is taken from the paper here: Let $L$ be an $n$-dimensional lattice. The Voronoi cell $V(u)$ is then defined as the set $\{x\in \mathbb R^n:|x|\le |x-v|, \mbox{ for all }v\in ...
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Prove (or disprove) that quantizing the end-points of a line segment does not change its quantized points

Let $q(v) := \lfloor v + 1/2 \rfloor$ denote the operation of quantizing a real number towards the nearest integer. Fix $x_0, x_1, y_0$, and $y_1$ where $x_i \in \mathbb{Z}$ and $y_i \in \mathbb{R}$...
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On a homogeneous Diophantine equation

I want to solve the diophantine equation $c_1x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ (I), when $c_1+c_2+c_3+c_4=0$. I first consider $x_1 = t$ and $(c_2+c_3+c_4)t = c_2x_2+ c_3 x_3 + c_4 x_4$; then I ...
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Getting an equivalent description of reflexive polytopes

In the proof of Theorem 4.6 from Computing the Continuous Discretely by Beck and Robins, the authors want to prove the equivalence of some descriptions of relfexive polytopes: $\mathcal{P}=\{\mathbf{x}...
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Non-parity based approach to the problem of showing that the area of a convex lattice pentagon is $\ge 5/2.$

I solved the problem given in the title with my approach shown below, but I was wondering whether there's any cleaner solution that does not rely on dividing the vertices of the polygon into the ...
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$(x,y)$ pairs in lattice $Z^2$ that are co-prime with euclidean-norm at most $k$

Let $B(k) = \{(x,y)\in Z^2 ~|~ x^2+y^2\leq k^2\}$, where $Z$ is the set of integers. It is quite straight forward to show that $|B(k)|$ is $\Theta(k^2)$. My question is whether the number of co-...
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Indecomposable elements in a lattice

Let $L$ be an discrete lattice in $\mathbb R^n$. We say that a nonzero $a\in L$ is indecomposable if and only if $a$ cannot be written as $a=b+c$ with $b,c$ nonzero and $b^T c>0$. I was initially ...
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Criteria for checking if points are the vertices of a hypercube

I asked a question over at Code Golf Stack Exchange which essentially asked folks to write a program to determine if a collection of $2^n$ points in $\mathbb{Z}^m$ is the vertex set of some $n$-...
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Complete Boolean Lattice

What is the formal definition of Complete Boolean lattice $Q_n$? What would be an example for $n=4$, i.e. $Q_4$?
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Points in a triangular lattice at the same distance from the origin and “breaking of symmetry”

Introduction I was trying to simulate what would happen to a certain physical system taking place in a triangular lattice (the physical details are not relevant to the discussion), when I came across ...
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Parametrization of all unimodular matrices of rank $n$?

For a given rank $n$, are there parametrization families that cover all possible unimodular matrices of rank $n$? For example for $n=3$ Wolfram gives an example of one parametrization family.
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Given a lattice supercell defined by three vectors, find the equivalent points in Wigner-Seitz Supercell

Assume you are given three primitive translation vectors for a lattice, $\mathbf{a}_1$, $\mathbf{a}_2$, $\mathbf{a}_3$. We construct a periodic supercell as all of the vectors $$\mathbf{r}(n_1,n_2,n_3)...
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When does a matrix have an “invariant quadratic form”?

Yesterday I computed that the matrix $$ A = \begin{pmatrix} 2&1\\1&1\end{pmatrix}$$ satisfies $q(m,n) = q \left((m,n)A\right)$ for the quadratic form $$q(m,n) = m^2 - mn - n^2.$$ E.g., $-1 ...
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Unusual action of $\text{SL}(2,\Bbb Z)$. Finding the invariant subspaces.

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\...
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Solving the HNP using lattice reduction?

I am trying to understand how lattice reduction can be used to solve the hidden number problem. The question came up after looking into this paper: https://eprint.iacr.org/2019/023.pdf The papers ...
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Testing for integer points in a parallelepiped

Given an invertible affine transformation $f:\Bbb R^n\to\Bbb R^n$ represented as $f(x)=Ax+b$, I'd like to know whether $f((0,1)^n)\cap\Bbb Z^n$ is nonempty. Is there an efficient way to compute this? ...
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Question about proof of theorem regarding cyclic sublattices

Suppose $L=[1,\tau]$ is an integer lattice with $\tau$ in the upper half-plane, $\gamma=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is an integer $2\times 2$ matrix with determinant $n$, and $...
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Computing the set of integral points of a convex hull

Assume that we have integral points $x_1, \ldots, x_n \in \lbrace 0, \ldots, l - 1 \rbrace^3$ for some $l \in \mathbb{N}_{> 0}$, that the vertices of the associated convex hull are given by $v_1, \...
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Determinant of lattice is $q^n$ with high probability?

Lemma: Let $\mathbf{A}\in\mathbb{Z}_q^{n\times m}$ be a uniformly random matrix and $\Lambda^\perp(\mathbf{A})=\lbrace x\in\mathbb{Z}^{m} : \mathbf{A}^Tx\equiv\mathbf{0}\ (\text{mod }q)\rbrace$ be a ...
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Relation between Weierstrass $\wp$-functions

Let $\Lambda=[\lambda_1,\lambda_2]$ be a lattice with associated Weierstrass function $\wp$, and consider the Weierstrass function $\wp_2$ associated to the lattice $\Lambda_2=[\tfrac{1}{2}\lambda_1,\...
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Block Positive Definite Quadratic Forms and their Representation Numbers

For a lattice $\Gamma=A\mathbb{Z}^n$ where $A$ is an $n$-dimensional invertible matrix, we define its squared length spectrum to be $L_\Gamma:=\{(\lambda^2,m):0\neq m=\{\gamma\in \Gamma: \lambda^2=\|\...
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Orthogonal complement to a lattice

Suppose we have an even lattice of rank 2, $\Lambda$, with the following intersection form, \begin{eqnarray} \left( \begin{array}{cc} 2 & 3 \\ 3 & 0 \end{array} \right) \end{eqnarray} As far ...
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helmholtz equation equipped with dirichlet boundary conditions for a square and the asymptotic behaviour of these corresponding eigenvalues.

Marcus here. The expression $-\nabla^2 u = \lambda u$, where $\nabla^2=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$, is know as helmholtz equation in two-dimensions. If this ...
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Is the sum of the reciprocals of the indices of sublattices in $\mathbb{Z}^{n}$ covering $\mathbb{Z}^{n}$ greater than or equal to 1?

For a finite number of sublattices $\Lambda_{i}\subseteq L:=\mathbb{Z}^{n}$, assume that $\cup_{i}\Lambda_{i}=L$. My guess is that $\sum_{i} \frac{1}{\left[L:\Lambda_{i}\right]}\ge 1$. Is my guess ...
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Exponentially decayed weight lattice path

Consider a 4 dimensional lattice with spacing $\Delta$. Want to get expression of $$\sum_\text{all paths} e^{-k(\Delta) L} $$ in the limit of $\Delta \rightarrow 0$ where $L$ is the number of "links"...
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Moving an object in a lattice

There's this problem I've been thinking about. Suppose we have a geometric shape (a ball, or a rectangle) of specific size, and we would like to know if it's possible to move it (by translating or ...
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Number of possible rotational domains of one 2D lattice on top of another?

There are four (or five) two dimensional Bravais lattices which I refer to as oblique, rectangular, hexagonal and square. I'll discuss the 17 symmetry wallpaper groups below. Ignoring translation, ...
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Finding shortest calculation of the sum of a subset of a group, given sums for other previously summed subsets

Say $S=\{g\in G\}$ is a set of elements in an abelian group $G$ whose group operation $(+)$ is expensive to compute. Given a subset $T\subset S$, we want to compute the sum of $T$'s elements, $\...
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How should we notate a D-dimensional lattice defined on multiples of natural numbers?

I have seen the lattice of natural numbers notated as $\mathrm{L}_\mathbb{N}$. It makes sense to me to write the $D$-dimensional lattice of natural numbers $\mathrm{L}_{\mathbb{N}^D}$. However, I’m ...
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Finding all lattice points in an $n$-dimensional hypercube

I have an $n$-dimensional axis-aligned hypercube and a lattice generated by $n$ linearly independent vectors, both in $\mathbb{Z}^n$. My wish is to find all points of the lattice which fall in the ...
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Distance in the plane [closed]

Chris labels every lattice point in the coordinate plane with the square of the distance from the point to the origin (a lattice point is a point such that both of its coordinates are integers). How ...
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Why for a maximal $G$ lattice of range $n$ holds that $Vol(G) = Vol(\mathbb{R}^n/G)$?

What I have is: Definition: A lattice $G$ is a subgroup of $V$ that is a real vector space on $\mathbb{R}$ of dimension $n$. A lattice is maximal iff $\mathbb{R}^n/G$ compact Th1: A lattice is ...
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Elements in a lattice with absolute value between two consecutive integers.

Let $\Gamma$ be a lattice in $\mathbb{C}$. I am currently reading the book "A course in arithmetic" from J.P Serre, and in page 83 (for the convergence of the Eisenstein series) he claims the ...
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Is there a standard name for the groups $SL(n,\mathbb Z)$ or $GL(n,\mathbb Z)$?

Is there a standard name for the set of all $n\times n$ integer matrices with determinant $1$ (special linear) or $\pm1$ (general linear)? The matrices themselves are called "unimodular", but ...
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Application of theorem on gluing vectors in lattice theory to $E_8$

I'm currently learning about gluing vectors in lattice theory, mainly from The (Sensual) Quadratic Form by Conway & Fung, and Sphere Packings, Lattices and Groups by Conway & Sloane. In the ...
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Closest Vector Problem in the 1-Norm. Mixed-Integer Linear Programming Formulation

So I understand the closest vector problem in the infinity-norm and getting to the final step of the program where I have: Minimize z Subject to z >= xj - tj for j = 1,2,...,n z >= tj - ...
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Understanding a crude estimate for the number of lattice points inside a ball

I've been reading the appendix (A lattice sum) of this write up by Keith Conrad and pretty much understand most of the argument but I'm stuck on the following: Let $S(x)$ denote the number of non-...
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On finding a finite set of generators for a certain semigroup

Let $A$ be a finite subset of $\mathbb Z^2$. Let $\mathbb ZA$ be the subgroup of $\mathbb Z^2$ generated by $A$. Let $\mathbb R_{+}, \mathbb Q_{+}$ denote the set of non-negative real and rational ...
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Equivalent of pick's theorem for a circle?

Consider the Pick's theorem that states that the area $A$ of a polygon placed on a grid of equal-distanced points such that all the polygon's vertices are grid points, is given by $A = i + \frac{b}{2}...
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General solution to this functional equation.

Let $\Lambda_8$ be the $E_8$ lattice. The functional equation is: $$f(a) = \sum\limits_{a=b+c} f(b)f(c)$$ Where $f(a)$ is defined only for points on the lattice $a\in \Lambda_8$ What would be a ...
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Sublattice of index 2

Let $M \subset L$ be two lattice of $\mathbb{R}^2$ and $|L:M|=2$. Let $v_1,v_2 $ be basis of $M$ and linearly independent in $L$. Assume further that $||v_1||\leq ||v_2||$. If we fix $v_1$, can we ...
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Covolume of discrete subgroups

Let $L \subset \mathbb R ^n$, and for every subgroup $L' \subset L$, denote by |L'| the covolume of L' in $\text{span}_{\mathbb R} (L')$. Prove the for every two subgroups $L_1, L_2$ of $L$: $...
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What are shifted lattices?

As a part of a project, I have been mentioned the work of deriving the functional equation of some $L$-functions associated with shifted lattices. Functional equations are not hard to obtain, but what ...
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45 views

Number of integral points in a polytope

I'm trying to count the number of integral solutions to the set of equations and inequalities \begin{equation} \begin{split} a_0 + a_1+a_2+a_3+a_4 = C \\ a_i \geq 0 \quad i \in\{0,1,2,3,4\} \\ a_0 \...
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Hints and help with 2011 AMC 10A Problem 25?

2011 AMC 10A Question #25 Let $R$ be a square region and $n\ge4$ an integer. A point $X$ in the interior of $R$ is called $n\text{-}ray$ partitional if there are $n$ rays emanating from $X$ that ...

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