Questions tagged [euclidean-lattices]

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Would anybody be able to summarise Minkowski Successive Minima in slightly simpler terms?

I came across this definition in Lenny Fukshansky's paper "Revisiting the hexagonal lattice: on optimal lattice circle packing" and I can't seem to fully grasp the concept of successive ...
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43 views

Under which assumptions on generating matrices $A,B$ is the lattice $A \mathbb Z^n$ contained in the lattice $B \mathbb Z^n$? [closed]

Let $A,B \in \mathbb R^{n \times n}$ be invertible matrices and let $L_1 = A \mathbb Z^n, L_2 = B \mathbb Z^n$ be the lattices in $\mathbb R^n$ generated by $A$ and $B$ (i.e. $L_1,L_2$ are the image ...
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2 votes
2 answers
78 views

The range of each of successive minima for all unimodular lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
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Embedding into odd unimodular lattices

Let $L$ be a unimodular even lattice of signature $(l+,l-)$, let $S$ be an even lattice of signature $(t+,t-)$, we know that when $(t+)+(t-)<\min(l+,l-)$, there exists a primitive embedding of $S$ ...
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1 vote
1 answer
111 views

Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes

Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+? Do you have any references explaining the relationships among these lattices and the 7D ...
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1 vote
1 answer
151 views

Standard Gram matrices for lattices

I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices. I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
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1 vote
0 answers
64 views

Intersecting lattices with surfaces in R^d

Let us fix some bounded surface $S\subset \mathbb{R}^d$. Let $x_1,\ldots, x_m$ be some non-zero vectors in $\mathbb{R}^d$. I am interested is the maximum number of points that the lattice $L_m=\{\sum ...
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3 votes
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66 views

How to determine sublattices S of a root lattice R

Let $R$ be a root lattice of a irreducible root system $\Phi$. Suppose $W$ is a Weyl group of $\Phi$ and $S$ is a sublattice of $R$ which is $W$-stable and satisfies $|R/S|<\infty$. For example, ...
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4 votes
1 answer
145 views

Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...
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0 votes
1 answer
250 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 ...
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Relation between integral lattice and an associated unimodular lattice construction

$M\in\mathbb Z^{n\times n}$ is an arbitrary integral matrix whose determinant is $det(M)$. We pick a random integral $T$ of size $O(1)\times O(1)$ having determinant $det(M)-1$. We construct using the ...
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On lattice generated by integral vectors close to eigenvectors

$M\in\mathbb Z^{n\times n}$ is an unimodular symmetric matrix whose eigenvectors are $v_1$ to $v_n$ and the lattice $\mathcal L$ generated by $M$ have the shortest vectors $u_1$ to $u_n$ where $u_i$ ...
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0 votes
0 answers
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On the shortest vectors of lattices generated by powers of generator matrices

Let $B\in\mathbb Z^{n\times n}$ generate a rank-$r$ lattice $\mathcal L_1\subseteq\mathbb Z^n$ and let $B^k\in\mathbb Z^{n\times n}$ generate lattice $\mathcal L_k\subseteq\mathbb Z^n$ assuming $\...
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3 votes
2 answers
310 views

Lattices containing $A_n$ and $D_n$

How many lattices are there which contain both the $A_n$ and $D_n$ lattices of the same dimension as sublattices? So far, I’ve found examples in 1D, 3D, 8D, and 24D.
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1 answer
115 views

Minimal volume of fundamental domains of lattices

Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one ...
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4 votes
0 answers
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Deep Holes of the tensor product of two lattices

Let $L_0, L_1$ be Euclidean lattices (say full rank) of dimension $n_i$. Let $\lambda_1(L_i)$ denote the length of the shortest vector of $L_i$, and let $\rho(L_i)$ denote the covering radius of $L_i$:...
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  • 702
7 votes
2 answers
595 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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  • 1,222
7 votes
3 answers
563 views

What other lattices are obtainable from this noncommutative ring?

Here I will regard $SU(2)$ as the multiplicative group of unit quaternions. There are just three conjugacy classes of finite subgroups $G < SU(2)$ where $[G:C] > 2$ for all cyclic subgroups $C &...
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3 votes
2 answers
141 views

Alternating 1D lattice sum

Are there any equivalent representations of the following (real valued) sum, in particular that are suitable for evaluation as $z\rightarrow0$ ? $$ S=\sum_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(...
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4 votes
1 answer
88 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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3 votes
0 answers
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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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  • 2,001
2 votes
0 answers
81 views

Shortest vectors in tensor product and maximal lattices in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
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4 votes
2 answers
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Can we count the number of integer lattice points in this case?

Gauss Circle problem gives the number of lattice points lie within a circle of radius $r$. This question points to a reference that estimates the number of lattice points in a $d−$dimensional ball. $...
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15 votes
0 answers
353 views

The Monster Moonshine Module from the engineering or algorithmic point of view

From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and ...
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10 votes
1 answer
341 views

Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not share an edge length or a vertex angle, and their areas differ. Suppose you'd like to tile the plane with pairwise incommensurate triangles. I ...
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5 votes
0 answers
200 views

Isomorphism classes of lattices

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
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8 votes
0 answers
94 views

Counting symmetric convex bodies with no nonzero lattice point in the interior

In order to estimate the size of the torsion in the algebraic $K$-groups of $\mathbb Q$ one needs to understand the homology of $\mathrm{GL}_n(\mathbb Z)$, or alternatively, the homology of the space ...
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3 votes
1 answer
147 views

Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$

One can define the convergence of a sequence $(\Lambda_k)_k$ of full rank lattices as folow : $(\Lambda_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists ...
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0 votes
1 answer
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Existence of linearly indépendants vectors reaching each minima of a lattice

I was wondering : given a full rank lattice $\Lambda$ of $R^n$ (a discrete subgroup spanning $R^n$) the successive minima of $\Lambda$ are for $1\leqslant i \leqslant n$ $\lambda_i= \min\{r>0 \mid \...
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  • 169
5 votes
0 answers
777 views

Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$: $$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
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3 votes
0 answers
39 views

Periodicity of density of laminated lattices

In Sphere Packings, Lattices and Groups, Conway and Sloane explore laminated lattices. If we let $X_d$ be the set of $d$-dimensional Euclidean lattices where every pair of points are separated by ...
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3 votes
0 answers
65 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 ...
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  • 1,009
4 votes
1 answer
133 views

On necessary condition for no integer points in polytope

For a convex polytope $\mathcal K$ in $\Bbb R^n$ presented by $O(n^c)$ linear inequalities is it true that for $|\mathcal K\cap \Bbb Z^n|=0$ it is necessary that at least one axis of John's ellipsoid ...
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2 votes
0 answers
112 views

Gap in Successive minima on lattice spanned by rational and integer combination of integer vectors

We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds. We have $$\mathcal L_\Bbb Z=\{uB\in\Bbb Z^n:u\in\Bbb Z^k\}\subseteq\mathcal L_\Bbb Q=\{uB\in\Bbb Z^n:u\in\...
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2 votes
2 answers
100 views

Integer points spanned by real, rational and integer combination of integer vectors

We are given a rank $r$ matrix $B\in\Bbb Z^{k\times n}$ where $0\leq r\leq k\leq n$ holds. We have $\mathcal L_\Bbb Z\subseteq \mathcal L_\Bbb Q\subseteq\mathcal L_\Bbb R$ where $$\mathcal L_\Bbb Z=\...
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1 vote
1 answer
126 views

Closest vertex in a 3D fcc lattice

The 3D fcc (face-centered-cubic) lattice, which has the same packing ratio as the 3D hexagonal close packed lattice, has the following 12 vectors connecting each vertex with its neighbors: $(1,-1,0),...
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3 votes
1 answer
170 views

"Sparse" Theta Series

The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function $$\theta_3(q)^2= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\...
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11 votes
1 answer
451 views

The lattice handshake number ("nearly kissing" number)?

Update: I'm happy to say that this question has been made essentially obsolete by the breakthrough result of Serge Vlăduţ, who showed that the kissing number is exponentially large: https://arxiv.org/...
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1 vote
1 answer
127 views

Densest sphere packing for a given lattice

There is a great deal of fascinating work on the sphere packing problem. I was wondering if there exist methods to find the densest possible packing of a sublattice of a given lattice. In particular, ...
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2 votes
0 answers
44 views

Bound on the fundamental region of a sublattice of a Niemeier Lattice

I am working with an 18 dimensional lattice, $W$ say, primitively embedded into a given Niemeier lattice $\mathcal{N}_i$. I am trying to figure out the following: what is the smallest possible ...
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6 votes
0 answers
204 views

Positive-definite lattice with O(n,n) Gram matrix generated by minimal vectors

Consider a positive-definite $2n$-dimensional lattice with minimum norm $\mu$. It is sometimes possible to find a generating set of minimal vectors for the lattice such that the Gram matrix takes the ...
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2 votes
0 answers
53 views

Configurations of minimal vectors for a 4-dimensional symplectic lattice

The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...
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1 vote
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On dimension of Segre embedding of lattice translations

Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$. Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\...
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0 votes
2 answers
274 views

Lattice question

Consider a lattice $\mathcal{L} = \mathbb{Z}v_1 \oplus \ldots \oplus \mathbb{Z}v_l$ in $\mathbb{R}^n$ and let $S_0$ be the set of edges of the fundamental unit of $\mathcal{L}$. We call a region $X$ ...
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  • 481
5 votes
0 answers
121 views

Lattice paths in polytopes

Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
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  • 481
5 votes
1 answer
173 views

lattice with Voronoi cell inside a circle

This considers real-valued lattices in two dimensions. I need to find the densest lattice $\Lambda$, i.e., the one with the smallest determinant of its generator matrix, such that the Voronoi cell of ...
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2 votes
2 answers
481 views

What is the spinor genus of the Leech lattice?

The Leech lattice and the 23 Niemeier lattices make a single genus. How does it break up into spinor genera?
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6 votes
0 answers
230 views

Bound on the determinant of a quadratic form restricted to a subspace

Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose: The minimum of $Q(x,y)$ as $y$ varies ...
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  • 5,758
2 votes
1 answer
233 views

Smallest angle among two lines in an n × n grid

Does anybody have a reference answering the following (at least for me surprisingly non trivial) question? Given an $n \times n$ integer grid, what is the minimum angle between any two distinct lines,...
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22 votes
1 answer
668 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
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