Questions tagged [euclidean-lattices]

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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 ...
Gro-Tsen's user avatar
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13 votes
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Multiplicity of ball covering

Background. My questions are motivated by the following: A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete and Computational Geometry, 8 (1992) 109-130) considered the ...
Misha's user avatar
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11 votes
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Lattices and stable homotopy groups of spheres

The number $65520$ arises in two very different scenarios: It occurs in the formula for the theta series of the Leech lattice: $$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...
Adam P. Goucher's user avatar
10 votes
0 answers
183 views

Boomerangs in Polya's orchard

Polya's orchard problem asks for what radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. The answer is known; $r$ ...
Joseph O'Rourke's user avatar
8 votes
0 answers
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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 ...
Xandi Tuni's user avatar
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7 votes
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Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.) Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.     $S(r_0,\epsilon)$ begins with the arc ...
Joseph O'Rourke's user avatar
6 votes
0 answers
226 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 ...
Chaitanya Murthy's user avatar
6 votes
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266 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 ...
Yoav Kallus's user avatar
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5 votes
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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 ...
Ramin's user avatar
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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{...
Jarek Duda's user avatar
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128 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 ...
Alex's user avatar
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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 $L_i$:...
Mark Schultz-Wu's user avatar
3 votes
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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, ...
Fuutorider's user avatar
3 votes
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53 views

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\...
Mircea's user avatar
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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 ...
Adam P. Goucher's user avatar
3 votes
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131 views

Enumerating 1-Lipschitz functions on an integer grid

Let $G$ denote an integer grid consisting of $\{0,\dots,m\}\times\{0,\dots,n\}$. An integer-valued function $f:G\to\mathbb{Z}$ is said to be 1-Lipschitz if it satisfies $|f(x) - f(y)| \leq \| x-y \|$...
Tom Solberg's user avatar
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3 votes
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Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
Campello's user avatar
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A non-commutative ring from SU(2)

$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 or 2. They are: ...
DavidLHarden's user avatar
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a counting problem on lattice

In $d$-dimensional lattice, we define a set $S_0$ be the zero point. At step $i\geq 1$: For each point $p\in S_{i-1}$, we can choose a single point $q$ who is a neighbour of $p$, and add $q$ into $s_{...
gondolf's user avatar
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2 votes
0 answers
170 views

Mistake in Rogers' paper: "number of lattice points in a set" for the case $n=2$?

Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in ...
taylor's user avatar
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Find the closest point of a lattice $\Lambda$ given the closest point for its union of cosets $\bigcup_i ({\bf r}_i+\Lambda)$

Suppose we have an $n$-dimensional lattice $\Lambda$, and a set of vectors ${\bf r}_i$, we can construct a union of cosets of $\Lambda$, denoted as $L$, as $$ L \equiv \bigcup_i ({\bf r}_i+\Lambda) $$ ...
fagd's user avatar
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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\...
Turbo's user avatar
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2 votes
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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\...
Turbo's user avatar
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2 votes
0 answers
51 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 ...
лаетековский чайник's user avatar
2 votes
0 answers
58 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 ...
crispus's user avatar
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Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
User133713's user avatar
1 vote
0 answers
29 views

Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?

Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$ $$ax+by=c.$$ Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
Turbo's user avatar
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1 vote
0 answers
42 views

How to construct lattices with largest possible number of Voronoi relevant lattice vectors?

Let M be the generator matrix of a $N$ dimensional lattice, and $V$ the set of Voronoi relevant vectors. The Voronoi cell for the origin can be written as $\text{Vor}_{\bf 0}(M)=\left\{{\bf x}: |{\bf ...
fagd's user avatar
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1 vote
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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 ...
TOM's user avatar
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1 vote
0 answers
103 views

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,\...
Turbo's user avatar
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1 vote
0 answers
135 views

A bounded function of the packing and covering density of lattices

Given a (finite-dimensional) lattice $L$ of an Euclidean vector-space, the function $$L\longmapsto -\log(\hbox{packing density of }L)/ \log(\hbox{covering density of }L)$$ is bounded and bounded away ...
Roland Bacher's user avatar
0 votes
0 answers
32 views

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 $\...
Turbo's user avatar
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0 votes
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Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$. (1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...
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0 votes
0 answers
882 views

Number of lattice points in a given triangle

Given a triangle with real coordinates, does anybody know how to find the number of lattice points contained within it? What if the points are only rational? I know Pick's formula can be used for the ...
Elliot Gorokhovsky's user avatar
0 votes
0 answers
245 views

upper bound on the size of sumset of lattice points

Let $\Lambda$ be a lattice (discrete additive subgroup) in $\mathbb R^n$ ($n\geq 2$). In my problem, $\Lambda$ lies in a $k$ dimensional ($1< k\leq n$) subspace of $\mathbb R^n$. Let $A\subset \...
EEstudent's user avatar
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0 answers
176 views

Quadratic forms and 0-1 points.

I have a quadratic form $Q(u) = \langle Du , u \rangle = 0$, where $D$ is circulant-symmetric from $\mathbb{R}^{n \times n}$ $D$ has all entries $0$ or $1$ except the diagonal which a negative real ...
Turbo's user avatar
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