Related papers: On Lattice Coverings by Simplices

G. Fejes T\'oth posed the following problem: Determine the infimum of the densities of the lattices of closed balls in $\bR^n$ such that each affine $k$-subspace $(0 \le k \le n-1)$ of $\bR^n$ intersects some ball of the lattice. We give a…

The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper to study {\it lattice coverings} in Nil space. We introduce the notion of…

In section 1 we give an improved lower bound on Hermite's constant $\delta_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. Here we…

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane.…

In this paper we prove a new asymptotic lower bound for the minimal number of simplices in simplicial dissections of $n$-dimensional cubes. In particular we show that the number of simplices in dissections of $n$-cubes without additional…

The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the…

This is a pedagogical review of the lattice study of finite density QCD, which is intended to provide the minimum necessary contents, so that the paper may be used as the first reading for a newcomer to the field and also for those working…

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…

In this paper I will approach the computation of the maximum density of regular lattices in large dimensions using a statistical mechanics approach. The starting point will be some theorems of Roger, which are virtually unknown in the…

It has been shown by Soprunov that the normalized mixed volume (minus one) of an $n$-tuple of $n$-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined…

We give bounds on the successive minima of an $o$-symmetric convex body under the restriction that the lattice points realizing the successive minima are not contained in a collection of forbidden sublattices. Our investigations extend…

Covering a graph or a lattice with non-overlapping dimers is a problem that has received considerable interest in areas such as discrete mathematics, statistical physics, chemistry and materials science. Yet, the problem of percolation on…

Diversities are a generalization of metric spaces, where instead of the non-negative function being defined on pairs of points, it is defined on arbitrary finite sets of points. Diversities have a well-developed theory. This includes the…

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…

It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the…

Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets…

It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we…

In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these…

In the dilute limit, the properties of fermionic lattice models with short-range attractive interactions converge to those of a dilute Fermi gas in continuum space. We investigate this connection using mean-field and we show that the…

For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first $\floor{d/2}$ dilations of P. As an application we give a necessary and sufficient…