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])
36 questions from the last 365 days
2
votes
0
answers
79
views
Question about lattice with dense projection
Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
- 305
3
votes
1
answer
132
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
3
votes
0
answers
63
views
Congruences regarding $4n$-dimensional lattices
A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if
$$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$
for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
- 111
5
votes
1
answer
162
views
I believe that all facets of a Voronoi-cell of a lattice are centerally symmetric. Is my argument correct? Is this true?
So let $L$ be a full dimensional lattice in $\mathbb{R}^{n}$. Then the Voronoi-cell of the lattice are precisely the points in $\mathbb{R}^{n}$ that are at least as close to the origin, as to any ...
1
vote
0
answers
51
views
Coarse-graining a hypergraph
$\DeclareMathOperator{\poly}{\mathrm{poly}}$I have asked this question on math.SE here, but couldn't get a satisfactory answer. I have also asked a related question on math overflow here, but haven't ...
- 277
7
votes
0
answers
150
views
Discriminants and lattices in Algebraic geometry vs Geometry of numbers
(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual ...
- 473
7
votes
1
answer
330
views
Is the partial order of all equations in the signature of magmas a lattice?
$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single ...
- 2,013
4
votes
1
answer
236
views
If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 43
55
votes
2
answers
3k
views
Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,053
1
vote
0
answers
52
views
Genus of binary quadratic forms: $f(x,y), g(x,y)$ in same genus if and only if represent same values in $(\mathbb Z/m\mathbb Z)^\ast$ for all $m$
In David Cox's book: Primes of the form $x^2+ny^2$, second edition, there is a theorem(Theorem 3.21, page 52) characterize whether two binary quadratic forms in the same genus. The contents of the ...
- 287
1
vote
0
answers
25
views
Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
2
votes
1
answer
95
views
What's the name of this constant similar to that of Hermite's?
Recently i've been thinking about base reduction of lattices, and this constant similar to Hermites constant came up. Let $L$ be a lattice with basis $\mathbf{b}_{1},\ldots,\mathbf{b}_{n}$. We define ...
1
vote
0
answers
44
views
Catalog of integral symmetric matrices
Let $g$ be an integral symmetric matrix (perhaps with even diagonal components), and define an equivalence relation $g\sim g'$ if $g=Ug'U^T$ with $U$ a unimodular integral matrix.
For fixed $\det g$ (...
3
votes
0
answers
61
views
For which lattices L does the cluster of Voronoi regions abutting that of the origin have a lattice tiling of euclidean space?
Let L be a n-dimensional lattice (a discrete cocompact subgroup of n-space).
Let V0 denote the Voronoi region of the origin, and let C denote the union of V0 with all the Voronoi regions that share a ...
- 2,858
3
votes
1
answer
106
views
Subgroups of a Weyl group fixing some vectors and its cohomology: MAGMA
I am trying to calculate the number of subgroups of the Weyl group $W(E_N)$ that fix certain vectors $L_i (i = 1,2,3)$ using Magma.
However, the output of the following code (especially #nicesubs) ...
- 1,364
1
vote
0
answers
66
views
Random lattice always has trivial automorphism group?
In example 2.5 of a paper [LS17] written by Lenstra and Silverberg, it is written that “Random” lattices have $Aut(L) = \{ \pm 1 \}$, I guess the 'Random' here refers to the distribution in Siegel's ...
- 253
2
votes
1
answer
123
views
Stabilizer of a lattice $\Gamma \subset G$ in the group of automorphisms $\operatorname{Aut}(G)$ is always discrete?
$\DeclareMathOperator\Aut{Aut}$Let $G$ be a (simply) connected Lie group whose semisimple part has no compact factors, and let $\Gamma $ be a lattice (uniform?) in $G$.
Is it true that the stabilizer $...
- 309
2
votes
0
answers
81
views
Lattice in a simply connected nilpotent Lie group
Given a connected and simply connected nilpotent Lie group $N$ with a left invariant metric, we assume that there is a lattice $\Gamma$ of $G$. Let $B_1(e)$ be the $1$-ball at the identity element in $...
- 21
12
votes
1
answer
380
views
Do lattices with small covering radius have sublattices with small covering radius?
For me a lattice is a discrete subgroup of $\mathbb R^n$. The linear span of a lattice, written $\Lambda \otimes \mathbb R$, is the $\mathbb R$-vector subspace of $\mathbb R^n$ generated by $\Lambda$. ...
- 148k
0
votes
1
answer
127
views
Automorphism groups in class sets of ternary lattices
Let $\Lambda$ be an integral lattice in some definite ternary quadratic space $(V,Q)$ over $\mathbb{Q}$.
Consider the usual class set $\text{Cl}(\Lambda) = O(V)\backslash\text{Gen}(\Lambda)$, i.e. the ...
- 562
1
vote
0
answers
66
views
Characterising lattices $\Lambda\subseteq\mathbb{Z}^n$ whose union of translations by $b\in\{0,1\}^n$ recovers $\mathbb{Z}^n$
Given a lattice $\Lambda\subseteq\mathbb{Z}^n$ defined by $\Lambda = \{ Mx : x\in\mathbb{Z}^n \}$, let $\Lambda_b$ for $b\in \{0,1\}^n = B$ be the translation of $\Lambda$ by $b$. Call $M$ special ...
- 11
3
votes
1
answer
399
views
Counting lattice points inside a parallelepiped
The problem I am about to state is in three dimensions and does not follow from Davenport's theorem. Its two-dimensional version is an immediate consequence of Pick's theorem.
Consider the lattice $\...
- 312
8
votes
1
answer
3k
views
Polynomial-time quantum algorithms for lattice problems (GapSVP, SIVP, LWE)
The author of a recent preprint claims to have found polynomial-time quantum algorithms for solving the following lattice problems: the Decisional Shortest Vector Problem (GapSVP), the Shortest ...
- 157
6
votes
1
answer
395
views
Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?
I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24:
Now let us recall that the ...
- 22.3k
3
votes
0
answers
65
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 ...
- 305
3
votes
1
answer
119
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 ...
- 31
0
votes
0
answers
32
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 ...
3
votes
1
answer
287
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?
- 13.9k
6
votes
1
answer
193
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 ...
- 163
1
vote
2
answers
276
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 $\mathbb Z^n$?
- 646
4
votes
0
answers
141
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 $n ≥ 3$...
- 911
2
votes
1
answer
131
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 ...
- 47
5
votes
1
answer
356
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 ...
- 460
0
votes
0
answers
64
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
81
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 ...
- 4,081
0
votes
0
answers
50
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. ...
- 1