New upper bound for lattice covering by spheres

Abstract

We show that there exists a lattice covering of ${\mathbb{R}}^n$ by Eucledian spheres of equal radius with density $O\left(n{\ln }^{2}n\right)$ as $n\to \infty$, where \begin{align*} β:= \frac{1}{2} \log_2 \left(\frac{8 π\mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O(n{\ln }^{\hat{I}}\pm n)$, where $\hat{I}\pm :=\frac{1}{2}{\log }_2(2\ddot{I}€\mathrm{e})=\mathrm{2.0471...}.$

15 pages