Bounds on the density of smooth lattice coverings

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.