New upper bound for lattice covering by spheres

2025/08/08 by Gao, Jun, Liu, Xizhi, Pikhurko, Oleg +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.06446

Abstract

We show that there exists a lattice covering of ℝⁿ by Eucledian spheres of equal radius with density O(n ln^β n ) as n→∞, where β:= (1)/(2) log₂ ((8 πe)/(3√ 3))=1.85837... . This improves upon the previously best known upper bound by Rogers from 1959 of O(n ln^α n ), where α:= (1)/(2) log₂(2πe)=2.0471... .

Citations

• Covering large-dimensional Euclidean spaces by random translates of a given convex body
• New bounds on the density of lattice coverings
• Successive Minima and Lattice Points

Cited by

• Nearly linear lattice coverings of arbitrary convex bodies

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