Questions tagged [euclidean-lattices]

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49 votes
4 answers
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What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
Joseph O'Rourke's user avatar
41 votes
2 answers
2k views

Can we find lattice polyhedra with faces of area 1,2,3,...?

I asked this question two months ago on MSE, where it earned the rare Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days. Perhaps justifiably so! Here I repeat it with ...
Joseph O'Rourke's user avatar
22 votes
1 answer
703 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,...
Theo Johnson-Freyd's user avatar
16 votes
2 answers
452 views

what is the equivalent of the Euler constant for higher dimensional lattices

Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$. Then there are constants such that $$\sum_{\substack{\gamma\in \Lambda\\0<|\gamma|<R\\}} \frac{1}{|\gamma|^d} = c_1 \log R + c_2 + o(1).$$...
Yannick Bonthonneau's user avatar
15 votes
0 answers
403 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 ...
Gro-Tsen's user avatar
  • 29.9k
13 votes
0 answers
562 views

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
  • 31k
11 votes
1 answer
541 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/...
Noah Stephens-Davidowitz's user avatar
11 votes
1 answer
481 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 ...
Joseph O'Rourke's user avatar
11 votes
0 answers
327 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{...
Adam P. Goucher's user avatar
10 votes
2 answers
790 views

Fitting a mesh to a density function

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
John Gunnar Carlsson's user avatar
10 votes
1 answer
616 views

A random variation on Polya's orchard problem

Polya's orchard problem is as follows: "How thick must the trunks of the trees in a regularly spaced circular orchard grow if they are to block completely the view from the center?" See, e.g., ...
Joseph O'Rourke's user avatar
10 votes
1 answer
358 views

Identifying lattices

I wrote a program that numerically searches for lattices in $\mathbb{R}^d$ with high sphere packing densities. As I have been running the program, it has been able to find, in addition to well-known ...
Yoav Kallus's user avatar
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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
9 votes
4 answers
930 views

Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 1) in mathematics 2) outside of mathematics, besides the applications to $...
8 votes
3 answers
616 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 &...
DavidLHarden's user avatar
  • 3,575
8 votes
1 answer
1k views

Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)?

Let $G$ be a $n\times n-$symmetric matrix with integral coefficients and determinant $1$ (i.e. unimodular) such that the associated quadratic form is positive-definite. I am interested in having an ...
Oblomov's user avatar
  • 2,501
8 votes
0 answers
99 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 ...
Xandi Tuni's user avatar
  • 3,975
7 votes
2 answers
828 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 ...
kaleidoscop's user avatar
  • 1,268
7 votes
2 answers
901 views

Is there a 3d equivalent of this picture?

This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin: Fitting a mesh to a density function The picture below is the image of a regular ...
John Gunnar Carlsson's user avatar
7 votes
1 answer
563 views

Upper bound for the number of integral points in a convex set

Let $K \subset \mathbb{R}^3$ be a bounded convex set such that the points with integer coordinates in $K$ are not all coplanar. Is it true that $|K \cap \mathbb{Z}^3| \leq 6{\rm Vol}(K) + 3$?
Slava's user avatar
  • 71
7 votes
0 answers
203 views

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
3 answers
921 views

Probability that a self-avoiding walk on $\mathbb{Z}^3$ closes to a polygon

The probability that a random walk on $\mathbb{Z}^3$ returns to the origin is about 34%. This is (part of) Pólya's theorem. I have been looking for an analogous (numerical) result for the probability ...
Joseph O'Rourke's user avatar
6 votes
2 answers
415 views

Bound on Minimal Length of Vectors in Lattice and its Dual Lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
Slava Rychkov's user avatar
6 votes
2 answers
302 views

Covolume of the row span of a matrix and of the kernel of a matrix

Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume $\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a fundamental domain for $L$, i.e., the volume of the parallelopiped ...
Joe Silverman'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
0 answers
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
  • 5,926
5 votes
2 answers
492 views

Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question. For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
Ariyan Javanpeykar's user avatar
5 votes
2 answers
276 views

Doubly covering an even lattice

I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
DavidLHarden's user avatar
  • 3,575
5 votes
1 answer
546 views

A 'generalized Four Squares Theorem'?

The $4$-dimensional lattice $\mathbb{Z}^{4}$ has vectors of length $\sqrt{n}$ for any positive integer $n$ by the Four Squares Theorem, but this need not be true for higher-dimensional integral, ...
DavidLHarden's user avatar
  • 3,575
5 votes
1 answer
183 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 ...
F Researcher's user avatar
5 votes
2 answers
391 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.
Daniel Sebald's user avatar
5 votes
0 answers
213 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 ...
Ramin's user avatar
  • 1,362
5 votes
0 answers
806 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{...
Jarek Duda's user avatar
5 votes
0 answers
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
  • 491
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. $...
Noah16's user avatar
  • 225
4 votes
1 answer
244 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 $\...
IMeasy's user avatar
  • 3,717
4 votes
1 answer
144 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 ...
Turbo's user avatar
  • 13.7k
4 votes
2 answers
492 views

Empty lattice simplex or White's theorem

White has proved (White, G. K. Lattice tetrahedra -- Canad. J. Math. 16 1964 389–396.) the following theorem: If $T$ is a closed tetrahedron and $\Lambda$ is a lattice which contains the vertices of $...
Alexey Ustinov's user avatar
4 votes
2 answers
174 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 ...
taylor's user avatar
  • 425
4 votes
1 answer
175 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 ...
Sean Miller's user avatar
4 votes
1 answer
106 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 ...
phdstud's user avatar
  • 133
4 votes
0 answers
125 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 $L_i$:...
Mark Schultz-Wu's user avatar
3 votes
2 answers
281 views

Extremal lattices

Denote by $\mu_n$ the largest value such that there exists a lattice of determinant $1$ in $\mathbb R^n$ for which the distances between different lattice points are greater or equal to $\mu_n$. ...
Portland's user avatar
  • 2,752
3 votes
1 answer
353 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 ...
Joseph O'Rourke's user avatar
3 votes
2 answers
183 views

Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$, and let $\beta(r)$ be the exterior buffer around $C(r)$: the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:   &...
Joseph O'Rourke's user avatar
3 votes
1 answer
213 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?
Turbo's user avatar
  • 13.7k
3 votes
2 answers
591 views

Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice. A set of integer vectors $v_1,\ldots,...
Campello's user avatar
3 votes
2 answers
188 views

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, ...
Joseph O'Rourke's user avatar
3 votes
1 answer
180 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}\...
Campello's user avatar
  • 800
3 votes
1 answer
241 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 $i$-...
taylor's user avatar
  • 425