Bounds on the density of smooth lattice coverings

TL;DR

This paper investigates the density bounds of smooth lattice coverings in high-dimensional spaces, demonstrating that for large convex bodies, such coverings exist with high probability, leveraging recent advances in the discrete Kakeya problem.

Contribution

It establishes asymptotic existence results for smooth lattice coverings of convex bodies with high probability, using novel techniques related to the discrete Kakeya problem.

Findings

Existence of smooth lattice coverings for convex bodies of volume n^{3+σ} as n→∞.

High probability of such coverings for random lattices under Haar-Siegel measure.

Results extend to random construction A lattices with polynomial bounds.

Abstract

Let $K$ be a convex body in $\mathbb{R}^n$, let $L$ be a lattice with covolume one, and let $\eta>0$. We say that $K$ and $L$ form an $\eta$-smooth cover if each point $x \in \mathbb{R}^n$ is covered by $(1 \pm \eta) vol(K)$ translates of $K$ by $L$. We prove that for any positive $\sigma, \eta$, asymptotically as $n \to \infty$, for any $K$ of volume $n^{3+\sigma}$, one can find a lattice $L$ for which $L, K$ form an $\eta$-smooth cover. Moreover, this property is satisfied with high probability for a lattice chosen randomly, according to the Haar-Siegel measure on the space of lattices. Similar results hold for random construction A lattices, albeit with a worse power law, provided the ratio between the covering and packing radii of $\mathbb{Z}^n$ with respect to $K$ is at most polynomial in $n$. Our proofs rely on a recent breakthrough by Dhar and Dvir on the discrete Kakeya problem.