New bounds on the density of lattice coverings

Ordentlich, Or; Regev, Oded; Weiss, Barak

doi:10.1090/jams/984

New bounds on the density of lattice coverings
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by Or Ordentlich, Oded Regev and Barak Weiss

J. Amer. Math. Soc. 35 (2022), 295-308

DOI: https://doi.org/10.1090/jams/984

Published electronically: July 28, 2021

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Abstract:

We obtain new upper bounds on the minimal density $\Theta _{n, \mathcal {K}}$ of lattice coverings of ${\mathbb {R}}^n$ by dilates of a convex body $\mathcal {K}$. 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 $L+\mathcal {K}= {\mathbb {R}}^n$. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem.

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