All Questions
Tagged with lattices reference-request
62 questions
55
votes
2
answers
3k
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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.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,063
1
vote
0
answers
25
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Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
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27
votes
7
answers
9k
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Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 111
3
votes
1
answer
119
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Seeking Article "Generating random lattices according to the invariant distribution" by M. Ajtai
I am searching for a specific article titled "Generating random lattices according to the invariant distribution" authored by Ajtai. Despite being widely cited in various papers, I have been ...
CommunityBot
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9
votes
1
answer
735
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 114k
2
votes
1
answer
166
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 ...
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5
votes
1
answer
2k
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Is there a relation between the number of lattice points lie within these circles
Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem.
Suppose that another ...
- 4,713
1
vote
0
answers
67
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Second moment version of the multiple-sum Rogers integration formula
I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure.
Theorem 1(Siegel-Rogers). Let ...
- 457
2
votes
1
answer
220
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Proof of generalized Siegel's mean value formula in geometry of numbers
Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
The classical Siegel's formula in geometry of numbers states ...
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3
votes
0
answers
86
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Sums over lattice points in homogeneously expanding domains
In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
- 3,107
0
votes
0
answers
81
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Extension of primitive set of vectors and reduction theory
Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$ (unimodularity is not really necessary here but just for convenience) and let $B$ be a ball centered at the origin that contains $(k+1)$-many $\...
- 457
2
votes
0
answers
171
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Bruhat-Tits tree as Cayley graph of free group
$\DeclareMathOperator\BT{BT}\DeclareMathOperator\GL{GL}$Let $p > 2$ be a prime and $n = \frac{p + 1}{2}$. We can identify the vertices of Bruhat-Tits tree $\BT(\mathbb Q_p)$ with the elements in ...
- 63.9k
5
votes
1
answer
672
views
coloring in lattice
This is a mathematical question raised from engineering and physics:
Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...
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1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 63.9k
2
votes
1
answer
76
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Reference request: Given a non-degenerate integral quadratic lattice $L,q$ over a PID, the quotient $L^*/L$ is given by SNF of $q$
Let $R$ be a PID with field of fraction $K$.
Let $L$ be a lattice with non-degenerate quadratic form $q:L\times L \to R$.
Let
$$
L^* = \{x \in L\otimes K \text{ s.t. } q(x,l) \in R \text{ for all } l \...
- 148k
2
votes
0
answers
85
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Showing an action of a higher rank lattice on hyperbolic space has a fixed point
In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
- 191
4
votes
0
answers
552
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Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
1
vote
2
answers
752
views
basis of the lattice generated by the integer points inside a subspace of R^L
Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{...
2
votes
0
answers
98
views
Sublattices in the standard integral symplectic lattice
Let $V$ denote $\mathbb{Z}^{2g}$ with its standard integral symplectic form $\omega = \sum_{i=0}^{g-1}dx_{2i} \wedge dx_{2i+1}$ (or, the homology lattice of a genus $g$ surface with its intersection ...
- 427
3
votes
1
answer
381
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Source on counting lattice points on a line
Looking for a book or article on the result linked below. The result tells us that the number of lattice points on a line between points $(a,b)$ and $(c,d)$ is given by $\gcd(a-c,b-d)+1$.
https://math....
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5
votes
1
answer
190
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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 ...
- 2,183
3
votes
1
answer
302
views
Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 25.7k
1
vote
1
answer
122
views
Property of convex polygons on integer lattice structures
Another graduate student and I are working on an research project and are looking for a paper or other source that has a proof for a result about polygons on an integer lattice structure. Suppose you ...
- 13.6k
6
votes
3
answers
855
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Fundamental solution of Discrete Laplace in the plane
We consider a discretization of the Laplace operator on $\mathbb Z^2$, https://en.wikipedia.org/wiki/Discrete_Laplace_operator
Then, it is natural to consider its fundamental solution $u$, i.e. $|u(x)...
CommunityBot
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2
votes
1
answer
112
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
...
- 5,529
24
votes
2
answers
889
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 ...
1
vote
1
answer
194
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$ ...
- 68.9k
12
votes
2
answers
980
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$...
- 2,339
6
votes
0
answers
550
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 ...
CommunityBot
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0
votes
0
answers
94
views
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 ...
- 3,209
1
vote
0
answers
522
views
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 ...
- 2,274
1
vote
1
answer
371
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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 ...
3
votes
1
answer
553
views
Lattice projections
I imagine the following result is folklore
Theorem. Those $k$-dimensional subspaces $\zeta \subset \mathbb{R}^n$ $(1 \leq k \leq n-1)$ for which the orthogonal projection of the lattice $\mathbb{Z}^n$...
CommunityBot
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1
vote
0
answers
278
views
Sphere packings with antipodal (unequal) spheres
Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
CommunityBot
- 1
1
vote
0
answers
99
views
Geometry of a $(d-1)$-dimensional lattice
Let $\mathbf u\in\mathbb Z^d$ be a primitive vector (i.e. $\gcd(u_i)=1$) and let $\Pi_{\mathbf u^\perp}$ be the orthogonal projection perpendicular to $\mathbf u$. I want to understand the geometry of ...
- 23.2k
1
vote
0
answers
53
views
Is it possible that a convex cone and its closure both induce vector lattices?
Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field.
Suppose that $P$ satisfies positive element stipulations.
(1) $X=P-P$.
(2) $P\cap-P=...
- 8,071
0
votes
0
answers
290
views
Need any information about an affine lattice
Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ ...
- 8,071
2
votes
1
answer
169
views
Higher dimensional analogs of logarithmic density
For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by
\begin{align*}
\underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\
\bar{d}(A)=\limsup_{N\to\...
- 104
2
votes
0
answers
78
views
automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space
Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...
- 975
3
votes
0
answers
259
views
Lattice points in regular simplex
Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
- 131
15
votes
1
answer
969
views
Counting lattice points inside a three-dimensional ellipsoid
I want to answer the following simple question:
Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^...
- 1,499
16
votes
1
answer
1k
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On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
- 43.7k
10
votes
0
answers
1k
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Bound on the number of lattice points in d-dimensional ball
The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
CommunityBot
- 1
4
votes
1
answer
203
views
Periodic functions over different lattices in $\mathbb R^d$ are linearly independent [closed]
I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim:
Let $f_1, \ldots, f_n$ be continuous ...
- 320
0
votes
1
answer
121
views
A limit of a sum related to integer lattice and power series
I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit
$$\lim_{z \to (1,\ldots,1)^-} (\sum_{v \...
2
votes
1
answer
131
views
Has the single sorted case of formal concept analysis been investigated?
A formal context in formal concept analysis is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects ...
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16
votes
2
answers
992
views
Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
1
vote
0
answers
86
views
Classification of involutions of the lattice $H\oplus H(k)^{\oplus2}$ for $k=5,6$?
Let $H$ denote the hyperbolic lattice (rank 2 lattice generated by $e,f$ such that $e^2=f^2=e.f-2=0$). Let $k >0$ be an integer. Is it possible to classify involutions $\iota$ of the lattice
$$
L:=...
- 1,663
9
votes
1
answer
946
views
Reference request: Ehrhart's conjecture on the geometry of numbers
Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n +...
CommunityBot
- 1
1
vote
1
answer
577
views
Minkowski's successive minima: A quantity not much larger than det(L)^(1/n) and not much smaller than λ_n(L)?
Let $\mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n$ be $n$ linearly independent vectors in an $n$-dimensional lattice $\Lambda$ in $\mathbf{R}^n$ and let $\mathbf{v}^*_1 ,\mathbf{v}^*_2, ..., \mathbf{...
- 30.5k