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])

Filter by
Sorted by
Tagged with
1
vote
0answers
71 views

Lattice points in hypercubes

Let $ (\Lambda_n) $ be a family of lattices, $ \Lambda_n \subset \mathbb{Z}^n $, with $ \det\Lambda_n \sim n $ as $ n \to \infty $ (meaning $ \lim_{n\to\infty} n^{-1} \det\Lambda_n = 1$). I am ...
1
vote
0answers
28 views

Finding a superbase in a lattice of Voronoi first kind

An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that $\{b_1,\ldots,b_n \}$ is a ...
3
votes
1answer
172 views

Number of points in a lattice and an oblong box

I have a very simple question in geometry of numbers. (It is a slight modification of Counting points on the intersection of a box and a lattice .) There's a bound I can easily prove, and it's good ...
6
votes
0answers
84 views

An integral Jacobson-Morozov theorem?

$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \...
2
votes
1answer
103 views

Reference request: placing a set with respect to the integer grid

For $x=(x_1,...,x_n)\in \mathbb{R}^n$, let $Q_x=(x_1,x_1+1)\times ...\times (x_n,x_n+1)$ - the open cube having $x$ in its "bottom left" corner. It seems, I can prove (see a draft here) the following ...
6
votes
2answers
500 views

what is the number of paths returning to 0 on the hexagonal lattice

I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice. I can find plenty on references on self avoiding paths, but I am looking ...
7
votes
2answers
250 views

Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$

Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial? The case where $G$ ...
0
votes
0answers
28 views

Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples - $\mathsf{II}$

$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Define the primitive Pythagorean triple $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. Consider the Linear Diophantine Equation $$a^{2t}u+b^{2t}...
0
votes
1answer
59 views

Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples

$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. Is it true that there are ...
0
votes
2answers
110 views

The union of two cuts is a cut?

Every poset $\langle P, \leq \rangle$ has a Dedekind-Macneille Completion, a complete lattice that embeds $\langle P, \leq \rangle$. For $A \subseteq P$, the upset $U(A) = \{p \in P\ |\ \forall a \in ...
0
votes
0answers
37 views

Upper bounds on the length of the shortest vector in lattices associated to polynomial congruences

We consider a lattice $\Lambda \subset \mathbb{Z}^2$, and put $\lambda_1(\Lambda), \lambda_2(\Lambda)$ for successive minima of the lattice $\Lambda$. By a well-known theorem of Minkowski, one has ...
2
votes
0answers
83 views

Another generalization of the Gauss circle problem

In this question I asked for a generalization of the Gauss circle problem, the type of generalization I am asking is to view the Gauss circle problem as one about counting algebraic integers of ...
6
votes
0answers
156 views

intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$. I have some intuition for $\mathbb{Z}$-lattices ...
0
votes
0answers
90 views

Definition of reducible lattice

I am reading Raghunathan's book on discrete subgroups of Lie groups. In particular I am stuck on Corollary 5.19 which gives several equivalent conditions for a lattice in a semisimple Lie group to be ...
5
votes
0answers
106 views

Lattices of minimal covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$

What are the (uniform/non-uniform) irreducible lattices of minimal (or even small) covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$? Context: Such a lattice will ...
5
votes
0answers
147 views

Number of rectangles in an n-by-n grid of points

I am seeking a formula for the number of rectangles with vertices belonging to an $n \times n$ square grid of points. This is sequence A085582 in the OEIS. Note that the grid in question has $n^2$ ...
4
votes
1answer
78 views

Closed cobounded additive submonoid of $\mathbb{R}^n$

Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more ...
1
vote
1answer
103 views

What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?

Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d_T$ induced ...
2
votes
1answer
122 views

Basis for a lattice in a subspace of $\Bbb R^n$

Let $S$ be a linear subspace of $\Bbb R^n$ having dimension $k<n$ and assume $S$ is described by $n-k$ linear equations with integer coefficients. Look at now the intersection $\Lambda=S\cap \Bbb Z^...
0
votes
0answers
88 views

Integers on Pentagrid

in de Bruijn's Pentagrid, the generating lines for Penrose Tiling are done by an intersection of a $2D$ hyperplane with the unit cube in $\mathbb{R}^5$, which formula can be found here: http://www....
3
votes
1answer
153 views

A problem of non-emptiness of intersections of certain chains of regular open sets

Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{...
24
votes
2answers
664 views

Simple conjecture about rational orthogonal matrices and lattices

The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ...
2
votes
1answer
119 views

Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$

Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not? If is an hard problem could give ...
4
votes
1answer
185 views

Unitary representations of lattices

Let $G$ be a simple linear group over a non-archimedean local field $F$. Assume that the split-rank over $F$ is at least 2. Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated ...
4
votes
0answers
76 views

Collections of points maximally spaced with respect to one another

The icosahedron and dodecahedron are well known to share symmetry groups. This partially accounts for the fact that one can form a type of compound of the two where each of the vertices in the ...
2
votes
0answers
110 views

Distribution of Smith normal forms for lower triangular matrices with given diagonals

Given integers $m$ and $n$ and $d_1, \ldots, d_m \in \mathbb{Z}/n \mathbb{Z}$, consider the set of all lower-triangular matrices of dimension $m$ with diagonal elements equal to $d_i$. What can be ...
4
votes
1answer
163 views

Free abelian subgroups of $\mathrm{SL}_n(\mathbb{Z})$

Does anybody know what is the biggest $r$ such that $\mathbb{Z}^r$ is isomorphic to a subgroup of $\mathrm{SL}_n(\mathbb{Z})$? It cannot be bigger that the virtual cohomological dimension of $\...
5
votes
0answers
99 views

Examples of non-uniform lattices in products of trees

Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete. So as a vague general question, what ...
2
votes
1answer
71 views

distribution of diagonal entries of Hermite Normal Forms

Consider a $n$-by-$n$ matrix $A$ over the integers and let $H$ be its Hermite Normal Form. Is there any result about the distribution of the diagonal entries of $H$, when $A$ is "randomly selected"? ...
1
vote
1answer
122 views

Reference request: The commensurator of an arithmetic lattice is a simple group

I am interested in a reference and proof for some version of the following (folklore?) statement: ``Let $G$ be a (semi)simple Lie group (with no compact factors and trivial centre) and let $\Gamma$ ...
7
votes
1answer
174 views

Is the commensurator of a tree lattice a simple group?

Let $T$ be an $n$-regular tree ($n\geq3$). Let $\operatorname{Aut}^+(T)$ be the subgroup of index 2 of $\operatorname{Aut}(T)$ preserving the bicoloring of the tree for which adjacent vertices have ...
0
votes
0answers
116 views

Is there exists a lattice isomorphism?

Let $\text{P}$ be the set of partitions of {1,2,...,n} and $\text{Y}$ the set of Young subsets of permutation group S(n)(the coxeter group of type An). As is well-known, the set $\text{Y}$ is a ...
11
votes
1answer
310 views

When does a locally symmetric space have no odd degree Betti numbers?

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\...
5
votes
1answer
228 views

The number of quadratic forms attaining Hermite's constant

$\require{AMScd}$ I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which ...
2
votes
2answers
125 views

enumerate line partitions of points in the plane

Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...
11
votes
2answers
782 views

Higman's lemma and a manuscript of Erdős and Rado

Motivated by a problem in factorization theory, I've recently proved the following: Theorem. If $X$ is a non-empty finite alphabet and $\mathcal W$ an infinite subset of the free semigroup, $X^\ast$...
5
votes
0answers
111 views

Lattices in Lie groups

In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups. Is there a result that gives a general description of a lattice in an arbitrary Lie group? Something ...
9
votes
1answer
482 views

Does every positive-definite integral lattice admit an angle-preserving homomorphism into $\Bbb Z^n$ for some $n$?

Some initial clarifications By lattice I mean an additive subgroup of $\mathbb R^n$ which is isomorphic to $\mathbb Z^n$ and has full rank (i.e. spans $\Bbb R^n$ when considered as set of vectors). A ...
5
votes
2answers
355 views

Lower bound for the number of lattice points on high dimensional spheres

Let $rS^{d-1}$ denote the sphere of radius $r$ in dimension $d$ (centered at the origin). I'm interested in the number of lattice points on the sphere (not inside). More precisely, let $$ N(r,d):=\...
4
votes
1answer
214 views

Counting points on the intersection of a box and a lattice

Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
1
vote
0answers
34 views

Lattices with no roots and spread out shells

I am looking for lattices with the following properties: The lattice has no roots. The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the ...
4
votes
0answers
77 views

Edges of the contact polytope of the Leech lattice

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$. Question: What are the edges of $P$? Let'...
8
votes
0answers
155 views

Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
1
vote
1answer
98 views

Submersion to $ T^{2}$

Let $ M$ be a $2n$-dimensional compact and connected manifold. Suppose there is $\Omega\in\Omega^{1}(M,\mathbb{C}) $ a closed complex form whose real and imaginary parts represent linearly ...
2
votes
0answers
36 views

Rational linear independence of holomorphic functions

Fix an integer $ m $ and a lattice $ \Lambda \subset \mathbb{C}^{m} $. Identify $ \Lambda \otimes \mathbb{R} $ with $ \mathbb{C}^{m} $. Take $ n $ holomorphic functions $ f_{1}, \ldots, f_{n}: U \to \...
21
votes
2answers
652 views

Which even lattices have a theta series with this property?

This is a slight generalization of a question I made in Math StackExchange, which is still unanswered after a month, so I decided to post it here. I am sorry in advance if it is inappropriate for this ...
1
vote
0answers
92 views

Lattices are not solvable in non-compact semisimple Lie groups

I'm trying to prove the following result. If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
2
votes
1answer
201 views

Find a lattice basis given too many points

Fix a discrete addition subgroup in $\mathbb{R}^n$. Given a finite spanning set, how can one find a group basis?
3
votes
1answer
133 views

Convex Hulls of Demazure Modules

Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\...
3
votes
0answers
49 views

Selfsimilar lattices in $\mathbb R^d$

Let $\Lambda\subset \mathbb R^d$ a discrete subgroup, up to diminishing $d$ we assume it is of the form $A\mathbb Z^d$ with $A\in GL(d)$. Up to dilation we assume that the shortest vector in $\Lambda\...

1
2 3 4 5
11