New bounds on the density of lattice coverings

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.