New bounds on the density of lattice coverings

TL;DR

This paper establishes new upper bounds on the density of lattice coverings in Euclidean space and analyzes the probability that random lattices cover space with convex bodies, advancing understanding of geometric lattice arrangements.

Contribution

It introduces improved bounds on lattice covering densities and probabilistic coverage measures, using strengthened results from the discrete Kakeya problem.

Findings

New upper bounds on lattice covering density

Probabilistic bounds on lattice coverage of space

Enhanced techniques from the discrete Kakeya problem

Abstract

We obtain new upper bounds on the minimal density of lattice coverings of Euclidean space by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies that L+K is all of space. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem.