Bounds on the density of smooth lattice coverings
Year: 2025
Type: article
Abstract: Let $$\cal{K}$$ K be a convex body in ℝ n , let L be a lattice with unit covolume, and let η > 0. We say that $$\cal{K}$$ K and L form an η-smooth cover if each point x ∈ ℝ n is covered by (1 ± η)vol( $$\cal{K}$$ K ) translates of $$\cal{K}$$ K by L. We prove that for any positive σ and η, asymptotically as n → ∞, for any $$\cal{K}$$ K of volume n 3+σ , one can find a lattice L for which $$\cal{K}$$ 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 ℤ n with respect to $$\cal{K}$$ K is at most polynomial in n. Our proofs rely on a recent breakthrough of Dhar and Dvir on the discrete Kakeya problem.
Source: Journal d Analyse Mathématique
Authors: Or Ordentlich, Oded Regev, Barak Weiss
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4408964862