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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How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
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Unimodular matrix to increase the minimum eigenvalue

Given a positive definite matrix $P$, I would like to find a unimodular matrix $U$ so that $U P U^T$ raises the minimum eigenvalue as much as possible. How can one find such a matrix $U$?
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Finding the odometer function of an abelian sandpile

As a computer scientist and "armchair" mathematician, I'm trying to replicate the images found here of Abelian sandpiles on a square lattice, where the initial configuration is $n$ chips on a single ...
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Adelic lattices

Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without ...
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Decomposition of quotient group of lattices

By the Chinese remainder theorem, we know that $\mathbb{Z}_m \cong \prod_{i=1}^l \mathbb{Z}_{p_i^{k_i}}$, where $m=p_1^{k_1} ... p_l^{k_l}$. Now, let $\Lambda = A(\mathbb{Z}^n) \subseteq \mathbb{Z}^...
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Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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Number of points in the diagonal of an $X \times Y$ square lattice box.

So assuming we have a $X \times Y$ lattice, Say for example a $3 \times 5$ like so * * * * * * * * * * * * * * * I need to find the number of points that each ...
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Nonplanar equilateral lattice “pentagons”

It is well-known that no two-dimensional point lattice contains a regular pentagon. (See for example http://mathworld.wolfram.com/LatticePolygon.html) The same is true for lattices in $\mathbb{R}^n$, ...
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Lattice Reduction Problem: Minimizing the “Longest” Basis Vector

Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec v_n)$....
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continued fractions and convex hulls

If I remember correctly, there is a nice correspondence between continued fractions and convex hulls of lattice points in the plane. If $\theta>0$ is the slope of a line in $\mathbb{R}^2$ passing ...
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The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
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An interesting connection between the Möbius function and the parity of the number of sublattices of index $n$ in generic $3$-dimensional lattice

I recently discovered an interesting connection between the following two On-Line Encyclopedia of Integer Sequences (OEIS) sequences: A001001 and A209635. More specifically, there seems to be an ...
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Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
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Expected time to walk in Manhattan

Suppose you are at coordinate $(m,n)$, and wish to go to $(0,0)$ using unit steps, either down or left. However, at each point, there is a traffic light, which switches from allowing horizontal to ...
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How can I visualize ideals on a ring of integers of imaginary quadratic fields?

If I were to visualize the ideal $(2, 3+3i)$ of $Z[i]$ on the complex plane, I would find a gcd of $2$ and $3+3i$ (for example $1+i$) and the ideal $(2, 3+3i)$ is identical to $(1+i)Z[i]$, which forms ...
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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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Leech Lattice and Golay Code

Consider the following Miracle Octat Generator or MOG. Choose the sign $\pm 3$ and fill in the blanks $\pm 1$ to create a point $x$ in the Leech lattice $\Lambda_{24}$ with $||x||^2=8$. $ \frac{1}{2}...
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$\Delta$ vs. $\delta$ in Lattice Theory

I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even ...
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lattice symmetries portrayed in animation

Below is an animated gif by Dave Whyte: What is the orbifold/fibrifold notation for the symmetries of the lattice depicted below? (a la /The Symmetries of Things/? Are there related theta function ...
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Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
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When does a simplex have an interior lattice point?

Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex ...
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Automorphisms of a lattice and changing to a nicer $\mathbb{Z}$-base

Suppose I have an integral lattice $L$ with an arbitrary $\mathbb{Z}$-base, equipped with a positive-definite nondegenerate symmetric bilinear form $\langle\cdot,\cdot\rangle$, and an isometry $\nu$ ...
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Term for intersection of lattice and convex region?

Is there a special term or convenient phrase for the restriction of a convex region to points of a lattice? This is motivated by wanting to talk about the feasible points of a discrete problem. I'd ...
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Probability distribution of a self avoiding walk

Preliminary: Consider a walk on the lattice $\mathbb{Z}_d$ lattice of length $N$. In a normal random walk, if we let $N$ get large the end position has a probability distribution (PDF) that looks ...
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Bijection between $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$ and lattices in $F^n$

I've come across mention of a bijection between lattices in $F^n$ ($F$ a field, in my case $\mathbb{C}(\!(t)\!)$) and elements of $\operatorname{GL}_n(F)/\operatorname{GL}_n(O)$, where $O$ is the ring ...
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Construction of a symplectic basis for a lattice

Let $(T,E)$ be a polarized abelian variety ($T=V/L$, $\dim_\mathbb{C} V=g$, $E:V\times V\to\mathbb{R}$ a nondegenerate real alternating bilinear form, with $E(L\times L)\subseteq\mathbb{Z}$ and $E(iv,...
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Why is $ \theta(m) \propto \zeta(2) $ if it is counting lattice points in a hyperbola?

I found this lattice point identity in a derivation of $\zeta(2)$: $$ \theta(x) = \sum_{mr \leq x} m = \sum_{r \leq x}\sum_{m=1}^{[x/r]} m = \sum_{r \leq x} \left( [x/r]^2 + [x/r] \right) = \sum_{...
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Intersection of two sets of rationals

I'm looking to see if anyone has any solutions or references for this problem. I'm not even sure of a proper category. It seems like it should be trivial, perhaps I'm missing something obvious. ...
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Conjecture about circles in plane lattices

A plane lattice $\Lambda$ is a set $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, where $A,B$ are linearly independent vectors in $\mathbb R^2$. The set of all circles in $\Lambda$ is $$\mathcal K(\...
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How interpret the dual lattice $\Gamma^*$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \...
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Lenstra's integer programming algorithm: Finding a lattice point “near the center”

Preliminaries: As part of Lenstra's algorithm for integer programming (see here, page 4) we compute a linear transformation $\tau$ and a point $z \in \mathbb{R}^n$ which meet certain conditions (step ...
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Index of a sublattice

Assume that $\Lambda \subset \mathbb{Z}^2$ is an integer lattice of rank $2$ and define for each integer $k$, $$\Lambda(k):=\Big\{(x_1,x_2) \in \Lambda: k\mid (x_1,x_2)\Big\}.$$ What is the index of $[...
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Creating n-dimensional lattices from lower dimensional parts

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite $...
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Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
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Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
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Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n $ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
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Coexistance of certain vectors in a lattice

I'm currently working on the SVP (Shortest Lattice Vector Problem) as a part of a paper that I'm writing. I've been trying to prove ( or disprove) the following without too much success : Question : ...
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Lower bounds on possible integer relations from the PSLQ algorithm

For the equation: $$ \sum_{i=1}^na_ix_i=0 $$ where all $x_i$ are real numbers and all $a_i$ are integers, the PSLQ algorithm can either find an integer relation or give lower bounds on the norm of ...
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Finding a basis for a particular integer lattice

The following problem arose in the context of string theory. I hope someone here might provide some guidance or a solution... Our starting point is: i) an integer-lattice $L\subset\mathbb R^n$ ...
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Line and distance to lattice points?

Consider the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ and a line (not necessarily through the origin). What conditions can be placed on the slope of the line that is necessary and sufficient so ...
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Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb C/\...
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Estimate for the co-volume of discs centered at lattice points in the plane?

Suppose I have a unimodular lattice $\Lambda = A \mathbb{Z^2}$ ($A\in SL(2,\mathbb{R})$) in the plane. I place a disc of fixed radius, $r$, around each point of $\Lambda$, so that I have a union of (...
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The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
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Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
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moduli of lattices

Consider the set $M$ of all (rank $g$) lattices in $g$-dimensional complex affine space $C^g$. Does M identify in some way with Siegel upper half space $H_g$? Let's say a lattice has CM if it has ...
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Which lattices are ideals of a number field?

Let $K$ be a number field, then its ring of integers $\mathcal{O}_K$ in the Minkowski space of $K$ is a lattice $\Lambda$. Is there some geometric descrpition/intuition that describes sublattices of $\...
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Non zero Unimodular matrices

I have a question about the lattice reduction algorithms like LLL algorithm. Lattice reduction algorithms like LLL generate a unimodular matrix which makes more orthogonal basises for a given matrix. ...
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Is the Voronoi region of a lattice symmetric around origion?

Assume an n-dimensional lattice. Is the Voronoi region of the lattice symmetric around origion? In other word, is the following statement true? "if $x\in \mathcal{V}$ then $-x\in \mathcal{V}$" where ...
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Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...