Questions tagged [lattices]
Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
652 questions
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Estimating an integral involving Bessel functions
I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
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Counting integral points on a surface
Let $f$ be a homogeneous polynomial with integral coefficients of 4 variables $a$, $b$, $c$ and $d$. Suppose $f$ is invariant under the rotation that rotates $(a,b)\in\mathbb{R}^2$ and $(c,d)\in\...
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Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
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Growth of products of large sets in Lie groups
The Steinhaus Theorem states that if $A$ is a set of positive measure in a locally compact group $G$, then the set $A *A^{-1} := \{ab^{-1} | a,b \in A \}$ contains a ball around the identity in $G$.
...
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Hamming weights of special vectors
The motivation of this question comes from number theory (I add the tag number theory for this reason, in that it is possible that someone with a number-theoretic background has already thought about ...
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What is the spinor genus of the Leech lattice?
The Leech lattice and the 23 Niemeier lattices make a single genus. How does it break up into spinor genera?
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Self-dual binary codes of Hamming weight divisible by 8?
Recall that a binary code is a subgroup $C \subset \mathbb F_2^n$; the elements of $C$ are called code words. The Hamming weight of a code word $c\in C$ is the number of $1$s in it. A binary code is ...
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Which unimodular matrices are integer combinations of permutations?
Consider the set of all integer linear combinations of permutation matrices of some fixed dimension. Is there a description of the set of unimodular matrices in this lattice?
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Intermediate lattices $C\mathbb{Z}^n \subseteq \Lambda \subseteq \mathbb{Z}^n$
Let $C \in \mathfrak{gl}(\mathbb{Z},n)$ be a symmetric full rank integer valued matrix (in my case it is the symmetric part of a Cartan matrix). Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank ...
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Is the genus symbol implemented?
Conway and Sloane, as well as Cassels, and also O'Meara, all have their own idiosyncratic way of expressing the following result (for good primes): Every quadratic form with coefficients in $\mathbb{...
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16-cell honeycomb (4D tiling by cross-polytopes)
A 4-dimensional cross-polytope (also called 16-cell) is a regular polytope whose vertices are all permutations of $(\pm1,0,0,0)$. It is known that this body tiles the space $\mathbb{R}^4$ by ...
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Solvability conditions for linear system of diophantine equations
Let $Ax=B$ be a system of linear diophantine equations, where $A$ is a full rank $n \times 2n$-matrix with integer entries. In the case $n=1$ we have solutions parameterized by $\mathbb{Z}$ iff $gcd(...
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shortest lattice point solution to a linear system
Consider the lattice $\mathbb{Z}^n$ and a real matrix $A\in \mathbb{R}^{m\times n}$ ($m<n$) with orthonormal rows. Let $y\in A\mathbb{Z}^n\setminus\{0\}$ and consider the equation $Ax=y$. Is there ...
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Bound on the determinant of a quadratic form restricted to a subspace
Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose:
The minimum of $Q(x,y)$ as $y$ varies ...
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Is this property a new large cardinal notion?
Given a cardinal $\kappa$, $\kappa$-complete lattices are lattices that have joins and meets of less than $\kappa$ elements (in particular they are bounded). In what follows we shall restrict to the ...
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Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
- 1,813
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Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
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Finding which Niemeier lattice a given basis corresponds to
This question is inspired by this post. A good answer to this question might help answer the linked post as well.
Question: Given a $24\times24$ matrix corresponding to an even, self-dual lattice in $...
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Properties of pairing-reversing automorphisms of integral lattices
Let $\Lambda$ be an integral lattice with the symmetric bilinear form (or the pairing) $\langle,\rangle$.
I'd like to know any reference (or even just the standard math terminology) for (the ...
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Intuition behind the definition of the Siegel-Eichler transformation
Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.
Let $X$ be an ...
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A question about smooth convex lattice polygons
Let $P$ be a smooth convex lattice polygon in $\mathbb{R}^2$ (the lattice being $\mathbb{Z}^2$). Here smooth means that at any vertex of $P$, the two primitive integer vectors (i.e. vectors whose ...
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Existence of lattices whose circles have bounded number of points
For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \...
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Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?
Let $k$ be a global field and let $S$ be a non-empty set of places containing all archimedean ones. Suppose $f:G\to H$ is a surjective $k$-morphism of $k$-groups and let $\Gamma\leq G(k)$ be an $S$-...
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Higher dimensional analogs of logarithmic density
For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by
\begin{align*}
\underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\
\bar{d}(A)=\limsup_{N\to\...
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The Linnik problem for dimension $2$
For $N$ an integer, let
$$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$
For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
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automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space
Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...
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Finding good high-dimensional sphere coverings in Euclidean space
Suppose we want to cover the unit sphere $\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ with spherical caps $\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...
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Lattice points in regular simplex
Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
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Lower bound for the number of representations of integers as sum of squares
Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...
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Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$
Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ ...
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Dual lattices up to a q scaling factor
In this paper : https://eprint.iacr.org/2011/501.pdf
There is an equality page 10, in the second paragraph considered by the authors as "easy to check". If someone could explain to me why the set at ...
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Understanding sphere packing in higher dimensions
In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved.
Admittedly it is very ...
user88381
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Existence of lattices in reductive Lie groups
What is known about existence of lattices in reductive Lie groups? The best results I know about existence of lattices in connected Lie groups are either about semisimple groups or nilpotent groups ...
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Counting lattice points inside a three-dimensional ellipsoid
I want to answer the following simple question:
Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^...
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Relation to Ehrhart polynomial with Uniqueness
A set of relative prime, positive integers $A = [a_1, \dots, a_d]$ describe the restricted partition function
$$
p_A(n) = \# \{(m_1,\dots,m_d)\in\mathbb{Z}^d: \textrm{ all }m_j \geq 0, \sum_{j=1}^d ...
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Uniform lattice in semidirect product
A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact.
My question is whether a uniform lattice exists in the group
$$
G={\mathbb R}^...
user1688
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What's in the genus of the cubic lattice?
I'll write $\mathbf{Z}^n$ for the integral quadratic form $x_1^2 + \cdots + x_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can ...
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Is there a nice choice-free argument to count the number of sublattices?
It's a well known fact that the number of index $n$ sublattices of a rank two lattice $\Lambda$ is given by $\sigma_1(n) = \sum_{d\mid n} d$.
Here is a proof of this fact:
Proof: choosing a basis of ...
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Lattice points near a curve
Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...
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Lenstra's integer programming algorithm: Finding a lattice point “near the center”
I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...
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On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
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Number of lattice points in homotetic image
I asked this question on MSE a week ago and it gave me a tumbleweed badge :-)
Let $\Lambda$ be a lattice in $\mathbb R^n$, with covolume $\Gamma$.
Moreover, let $S$ be a bounded (Lebesgue-)measurable ...
user40023
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Almost invariance in compact quotients of locally compact groups
While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups
in the topological setting, I face the following problem.
Let $G$ be a locally compact amenable ...
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Closest point to a dual lattice point (in particular for root lattices!)
Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
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Existence of the double coset ring on paper of Ihara
In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from $...
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Two types of random walkers on square lattice
Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index $1$...
user82424
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Fundamental solution of Discrete Laplace in the plane
We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
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Average of Short Character Sum over All Dirichlet Characters Mod n
Cross-posted from M.SE.
Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$.
I want to find an order-of-...
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Deformations of null-vectors of an integral unimodular lattice
Is there an $SO(n,n)$ transformation that makes the Euclidean norm-squared of all the null vectors of the $(n,n)$ hypercubic lattice strictly greater than 4?
Example: Let $\Lambda_{\rm sL}$ denote ...
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Question on some coverings of the euclidean space
Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
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