Lower Bounds on Lattice Covering Densities of Simplices

TL;DR

This paper establishes new lower bounds on the lattice covering densities of simplices, specifically tetrahedra and four-dimensional simplices, using the Degree-Diameter Problem for abelian Cayley digraphs.

Contribution

It introduces novel lower bounds for lattice covering densities of simplices by linking them to the Degree-Diameter Problem in abelian Cayley digraphs.

Findings

Lattice covering density of a tetrahedron is at least 25/18.

Lattice covering density of a 4D simplex is at least 343/264.

Provides theoretical bounds for lattice coverings of simplices.

Abstract

This paper presents new lower bounds for the lattice covering densities of simplices by studying the Degree-Diameter Problem for abelian Cayley digraphs. In particular, it proves that the density of any lattice covering of a tetrahedron is at least $25/18$ and the density of any lattice covering of a four-dimensional simplex is at least $343/264$.