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New bounds on the density of lattice coverings (2006.00340v1)
Published 30 May 2020 in math.NT, cs.IT, math.IT, and math.MG
Abstract: We obtain new upper bounds on the minimal density of lattice coverings of Euclidean space by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies that L+K is all of space. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem.