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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6 votes
2 answers
559 views

Counting points on lattices in inside a box- Geometry of numbers

Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and let $|\mathbf{x}|$ denote the $L^2$ norm. There is a fairly standard argument involving successive minima to obtain an estimate on $N(R)$ which is the ...
2 votes
1 answer
65 views

Seeking Article "Generating random lattices according to the invariant distribution" by M. Ajtai

I am searching for a specific article titled "Generating random lattices according to the invariant distribution" authored by Ajtai. Despite being widely cited in various papers, I have been ...
3 votes
1 answer
183 views

Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix

Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...
3 votes
0 answers
45 views

Arithmetic lattices are finitely presented

In the book "Kazhdan's Property (T)" by Bekka-de la Harpe-Valette, the following is stated on p.6 of the introduction: "Of course, it is classical that arithmetic lattices are finitely ...
0 votes
0 answers
26 views

Integral representation of completely alternating homogeneous functionals on semi-lattice of continuous functions

For a long time I've been interested in G. Choquet seminal work "Theory of capacities" (Annales de l’institut Fourier, tome 5 (1954), p. 131-295). More precisely part 53 about integral ...
6 votes
1 answer
182 views

What is the smallest $\mathbb{Z}[x]$ multiple of $(x-1)^n$ in the coefficient vector $\ell_1$ sense?

Define a norm $\lVert p \rVert_1$ for $p\in \mathbb{Z}[x]$ as the sum of absolute values of the coefficients of $p$, as expressed in the ordinary monomial basis. What is the smallest norm of a ...
3 votes
1 answer
212 views

On shortest vector problem

Assume we have an oracle which gives the length of the shortest vector in a lattice. Given this oracle can we find the shortest vector in polynomial time?
1 vote
2 answers
262 views

Does $\mathbb Z^n$ contain $A_n$?

Are there any positive integer $n > 3$ such that the root lattice $A_n$ is contained in $\mathbb Z^n$?
22 votes
4 answers
2k views

What exactly is the relationship between codes over finite fields and Euclidean sphere-packings?

So I know that error-correcting codes are sphere packings in the Hamming metric, and that intuition and technical tools from the Euclidean case can often be applied to the finite-field case and vice ...
4 votes
0 answers
134 views

Vanishing of $\ell^2$-Betti numbers of $\mathrm{GL}(n,\mathbb{Z})$ for $n\geq 3$

$\DeclareMathOperator\GL{GL}$In a paper I read the following claim: By the work of Borel the $\ell^2$-Betti numbers of the cocompact lattices of $\GL(n,\mathbb{R})$ are known to all vanish when $n ≥ 3$...
2 votes
1 answer
111 views

Kripke frame, lattice and some intermediate logics

For a given finite and rooted intuitionistic Kripke frame $\mathcal{F}$, let $\log(\mathcal{F})=\{\phi : \mathcal{F}\vDash \phi\}$ and assume $S=\{\log(\mathcal{F}): \mathcal{F} \text{ is finite and ...
1 vote
1 answer
323 views

Lattices and noncommutative algebras in noncommutative geometry

This a question that I've asked in mathematics stack exchange without having received any response : I am interested in the relation between lattices and noncommutative algebras in the context of ...
5 votes
1 answer
320 views

Lattice generated by parabolics

Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free. For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
16 votes
4 answers
580 views

The lattice spanned by $m$ random 0-1 vectors of length $n$

Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
7 votes
1 answer
499 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\...
0 votes
0 answers
58 views

What is the lattice point distribution over binary quadratic forms?

Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$. For simplicity, we keep things only on quadrant I of the ...
0 votes
0 answers
78 views

Lattice not contained in any connected subgroup is not contained in any positive dimensional subgroup

Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any ...
0 votes
0 answers
45 views

Kirszbraun-like extension of periodic functions

Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
0 votes
0 answers
115 views

Roots in indefinite lattice of K3 surfaces

Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$). Inside we have ...
5 votes
2 answers
609 views

A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$. ...
6 votes
1 answer
225 views

Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?

Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
2 votes
0 answers
97 views

Efficient decoding of the E8/Leech lattice

Background: Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....
1 vote
0 answers
96 views

Number of points in a ball in positive characteristic

Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation. Assume that $w_1,...
4 votes
0 answers
200 views

Automorphism group of a Lorentzian lattice

Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product $$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$ Its ...
0 votes
1 answer
119 views

How common is it that the number of the shortest vectors in a lattice is exactly two?

The lattice $\Gamma$ in $\mathbf{R}^{m}$ with the lattice basis $\{ke_{k}\}_{k=1}^{m}$ has exactly two shortest vectors: $\pm e_{1}$. My question is the following: Among all the lattices with fixed ...
5 votes
1 answer
124 views

Expected value of the length of the shortest non-zero vector in a lattice?

$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
2 votes
0 answers
154 views

Will Coppersmith's method work for this bivariate modular polynomial shape?

I have a bivariate modular polynomial of shape $$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$ where $q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$, $g(x)\in\mathbb Z[x]$ is of degree four and $f(...
9 votes
1 answer
700 views

Where has this structure been observed?

$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure: $R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation": $$R_X (x, y) \cdot R_Y (x +...
13 votes
2 answers
2k views

Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and $\...
4 votes
0 answers
109 views

Advice on results for balls on regular $N$-dimensional grids

I have obtained some results regarding balls on regular $N$-dimensional grids. I would like expert opinion on wether the results are significant or interesting enough for (trying to) publish them in a ...
1 vote
0 answers
55 views

Condition on the minimality of Minkowski units

I am interested in to undrestand when the Minkowski units in real biquadratic number fields are minimal in the log unit lattices. I have read some pieces of literature online which are investigating ...
4 votes
1 answer
697 views

Abelian subfactors, a relevant concept?

Through the questions below, this post asks whether the concept of abelian subfactor is relevant. Remark : here abelian qualifies an inclusion of II$_1$ factors $(N \subset M)$, $N$ is not an abelian ...
3 votes
0 answers
110 views

Characterizing the D4 lattice as a sphere packing

Suppose I pack spheres in $\mathbb{R}^4$ in such a way that each touches 24 others. (All spheres in my question are assumed to have equal radius and be non-overlapping.) Does this packing ...
2 votes
1 answer
375 views

Tri-coloring of E8 lattice? Why is the Gram matrix of E8 not unique?

This question is about euclidean lattices, regular arrays of points in $\mathbb R^N$. Why are there 3 Gram matrices for the E8 lattice? They are not related by a similarity transformation; they have ...
2 votes
3 answers
1k views

Countable atomless boolean algebra covered by a larger boolean algebra

Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
2 votes
1 answer
158 views

About finitely generated lattices in Lie groups

Let $G$ be a connected Lie group. Let $\Gamma$ a lattice in $G$ not necessarily uniform (cocompact). Is it true that $\Gamma$ is finitely generated?
3 votes
0 answers
85 views

Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?

Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form \begin{equation} \begin{pmatrix} 0 & \Delta \\ -\Delta & 0 \end{pmatrix}, \quad ...
25 votes
1 answer
1k views

How random are unit lattices in number fields?

I was wondering how random unit lattices in number fields are. To make this more precise: If $K$ is a number field with embeddings $\sigma_1, \dots, \sigma_n, \overline{\sigma_{r+1}}, \dots, \...
6 votes
0 answers
227 views

What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
0 votes
0 answers
112 views

Genus of quadratic form

I am trying to understand the genus of a lattice from Conway and Sloane textbook. They said two quadratic forms $Q_1$ and $Q_2$ lie in the same genus if they are equivalent over $\mathbb{R}$ and over ...
3 votes
1 answer
90 views

Lattice basis reduction over rings of number fields

Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean ...
3 votes
1 answer
138 views

Cohomology of cocompact lattices in hyperbolic spaces

I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
4 votes
2 answers
190 views

Constructions of complex surfaces covered by the ball of $\mathbb{C}^2$

Let $S$ be a compact complex surface. It is well-known that the following two facts are equivalent $c_1^2(S) = 3 c_2(S)$ and $S \neq \mathbb{CP}^2$ The universal cover of $S$ is biholomorphic to the ...
3 votes
0 answers
122 views

group generated by unipotents in arithmetic subgroup is finitely generated

Let $G$ be a semisimple algebraic $\mathbb{Q}$-group and $\Gamma$ an arithmetic subgroup of $G$. In particular $\Gamma$ is finitely generated. Denote by $\Gamma^{u}$ the set of unipotent elements in $\...
2 votes
1 answer
176 views

The sum of $q^{-2}$ over nonzero Gaussian integers

I'm reading about the Weierstrass zeta function. In this context, $\phi(z)=\zeta(z)-\pi\bar{z}$ is periodic over the lattice $$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$ If we take $w\in\mathcal{L}\...
7 votes
1 answer
322 views

Extremal problem for 2-dimensional lattices

Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly ...
5 votes
0 answers
139 views

Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
6 votes
1 answer
197 views

Preserve validity between the two Kripke frames

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame. For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...
2 votes
1 answer
146 views

Cocompact lattices in $\mathrm{Sp}(n, 1)$

This is a continuation from my previous question. I am reading the following paper of Cowling-Haagerup, and I was wondering whether there are uniform lattices in $\mathrm{Sp}(n, 1)$. Is there some way ...
4 votes
0 answers
105 views

Questions related to a paper of Cowling-Haagerup and uniform lattices of $\mathrm{Sp}(1,n)$

I am reading the following paper of Cowling and Haagerup for my master’s thesis. I am new to this area so I am not very conversant. So I do apologize if the questions are silly. Question 1. In the ...

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