# 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])

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### 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 ...

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### 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 ...

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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 ...

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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 \...

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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
...

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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 ...

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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 $G$ ...

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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}...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $n^2$ ...

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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 ...

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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 ...

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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^...

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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....

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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{...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $\...

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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 ...

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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"? ...

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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 $\Gamma$ ...

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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 ...

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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 ...

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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}:=\...

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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 ...

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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 ...

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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, $X^\ast$...

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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 ...

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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 ...

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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):=\...

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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\...

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### 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 ...

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### 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'...

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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 ...

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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 ...

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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 \...

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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 ...

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### 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 ...

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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?

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### 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_{\...

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### 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\...