New upper bound for lattice covering by spheres

TL;DR

This paper establishes a new upper bound on the density of lattice coverings of Euclidean space by spheres, improving previous bounds and advancing understanding of sphere covering efficiency in high dimensions.

Contribution

It introduces a tighter upper bound on lattice covering density, reducing the exponent in the logarithmic term compared to prior results.

Findings

New upper bound of $O(n \, \ln^{1.858} n)$ for lattice sphere coverings.

Improvement over Rogers' 1959 bound with a smaller logarithmic exponent.

Advances theoretical understanding of high-dimensional lattice coverings.

Abstract

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^{\beta} n \big)$ as $n\to\infty$, where \begin{align*} \beta := \frac{1}{2} \log_2 \left(\frac{8 \pi \mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^{\alpha} n \big)$, where $\alpha := \frac{1}{2} \log_{2}(2\pi \mathrm{e})=2.0471...\,.$