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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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\...
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Relating the shortest vector of a lattice to the orthogonal complement of the lattice
By a lattice we mean sub-lattice of $\mathbb{Z}^n \cap V$, where $V$ is a subspace of $\mathbb{R}^n$ defined over $\mathbb{Q}$. We say that a lattice $\Lambda$ is primitive if a basis of $\Lambda$ can ...
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Partitioning $\{0,1\}^n$ into $n$ sets
I am working on an answer to the question
Magic trick based on deep mathematics
and came across the following problem: I am trying to partition the cube $\{0,1\}^n$ into $n$ sets $P_1,\dots,P_n$ ...
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Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients
Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:
$$(i)\ 1 + 80q^3 ...
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Constructive proof of Swan theorem
Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably ...
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List of Automorphism groups of Abelian Varieties for Dummies
(%Edited after abx comment%)
I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit ...
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An interesting sum over lattice points in a large disk centered at the origin
Evaluate the the limit, as $r \rightarrow \infty $, of the sum $\displaystyle \sum \limits_{(m,n) \in D_r}$ $\displaystyle (-1)^{m+n} \over \displaystyle m^2 + n^2$, where $D_r$ denotes the closed ...
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Smallest integer lattice point by box measure in a polytope?
Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
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Lowering $i$th shortest vector of a lattice
LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with
$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.
...
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Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
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Which lattices are rotatable into their scaled copy?
Let $L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $d$ independent vectors $(v_i)_1^d$ from $\mathbb R^d$. Call $L$ rotatable if for some $M$, a scalar multiple of some ...
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Shortest Lattice Vector with restricted $x$
Let $\Lambda$ be a lattice with basis, $B$ consisting of vectors $b_i$, so that the elements of $\Lambda$ are of form, $y\in \Lambda \iff y=Bx=\sum_i b_ix_i$ for some $x_i\in\mathbb{Z}$.
My questions ...
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Basis of cone lattice
I only want to know whether a construction that I use appears in literature and maybe has a name already.
Let $V$ be a $\mathbb Q$ vector space of dimension $d\in\mathbb N$.
A subset $C\subset V$ is ...
user130903