Let \(\cal{K}\) be a convex body in ℝn, let L be a lattice with unit covolume, and let η > 0. We say that \(\cal{K}\) and L form an η-smooth cover if each point x ∈ ℝn is covered by (1 ± η)vol( \(\cal{K}\) ) translates of \(\cal{K}\) by L. We prove that for any positive σ and η, asymptotically as n → ∞, for any \(\cal{K}\) of volume n3+σ, one can find a lattice L for which \(\cal{K}\) , 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 \(\cal{K}\) is at most polynomial in n. Our proofs rely on a recent breakthrough of Dhar and Dvir on the discrete Kakeya problem.
