Bounds on the density of smooth lattice coverings

Let K be a convex body in ℝⁿ, let L be a lattice with unit covolume, and let η > 0. We say that K and L form an η-smooth cover if each point x ∈ ℝⁿ is covered by (1 ± η)vol(K) translates of K by L. We prove that for any positive σ and η, asymptotically as n → ∞, for any K of volume n^3+σ, one can find a lattice L for which K, L form an η-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 that the ratio between the covering and packing radii of ℤⁿ with respect to K is at most polynomial in n. Our proofs rely on a recent breakthrough of Dhar and Dvir on the discrete Kakeya problem.