# Questions tagged [euclidean-lattices]

The euclidean-lattices tag has no usage guidance.

95
questions

3
votes

2
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### Random walk to visible lattice points

Consider a random walk from the $\mathbb{Z}^2$ origin $(0,0)$ to visible
(not blocked) lattice
points $p$, with a parameter $r$ a given radius of a circle centered
on $p$.
With $p$ the previous point, ...

3
votes

1
answer

320
views

### Illumination from visible lattice points with inverse square intensity

It is well known that the number of $\mathbb{Z}^2$ lattice points visible from
the origin is $6/\pi^2$, about $61$%.
See, e.g.,
What fraction of the integer lattice can be seen from the origin?.
I am ...

1
vote

0
answers

28
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 $NC$ ...

1
vote

1
answer

74
views

### Existence of some lattice path connecting all given lattice paths

My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...

2
votes

0
answers

85
views

### 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_{...

2
votes

0
answers

155
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 ...

4
votes

2
answers

157
views

### How large is the set of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice?

Recall that the $i$-th successive minimum of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...

2
votes

1
answer

178
views

### 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 ...

3
votes

1
answer

179
views

### Successive minima and the basis of lattice

I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...

2
votes

0
answers

93
views

### 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
$$
L \equiv \bigcup_i ({\bf r}_i+\Lambda)
$$
...

1
vote

0
answers

37
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 ...

4
votes

1
answer

238
views

### Relation between positive roots of $E_8$ and $\mathbb{F}_2^8 \setminus \{0\}$

There exists an explicit bijection (due to Cayley, that has built up a very nice table to describe this) between the positive roots of the lattice $E_7$ and $\mathbb{F}_2^6 \setminus \{0\}$ (where $\...

11
votes

0
answers

310
views

### 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{...

2
votes

2
answers

121
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 ...

1
vote

1
answer

123
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 ...

0
votes

1
answer

267
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 ...

1
vote

0
answers

70
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 ...

3
votes

0
answers

81
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, ...

4
votes

1
answer

170
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 ...

1
vote

1
answer

360
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 ...

0
votes

0
answers

30
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 $\...

4
votes

2
answers

360
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.

1
vote

1
answer

146
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 ...

4
votes

0
answers

115
views

### 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 $L_i$:...

7
votes

2
answers

785
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 ...

8
votes

3
answers

602
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 &...

3
votes

2
answers

151
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+(...

4
votes

1
answer

97
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 ...

3
votes

0
answers

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\...

2
votes

0
answers

84
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\...

4
votes

2
answers

2k
views

### 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.
$...

15
votes

0
answers

390
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 ...

11
votes

1
answer

463
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 ...

5
votes

0
answers

208
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 ...

8
votes

0
answers

98
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 ...

3
votes

1
answer

158
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 ...

1
vote

1
answer

93
views

### 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 \...

5
votes

0
answers

796
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{...

3
votes

0
answers

42
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 ...

3
votes

1
answer

89
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 ...

4
votes

1
answer

140
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 ...

2
votes

0
answers

118
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\...

2
votes

2
answers

102
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=\...

1
vote

1
answer

146
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),...

3
votes

1
answer

177
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}\...

11
votes

1
answer

497
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/...

1
vote

1
answer

145
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, ...

2
votes

0
answers

50
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 ...

6
votes

0
answers

219
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 ...

2
votes

0
answers

55
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### 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 ...