On the covering multiplicity of lattices

Abstract

Let the lattice Λ have covering radius R, so that closed balls of radius R around the lattice points just cover the space. The covering multiplicity CM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for an n-dimensional lattice is n if n≤8, and conjecture that it exceeds n in all other cases. We determine the covering multiplicity of the Leech lattice and of the lattices In, An, Dn, En and their duals for small values of n. Although it appears that CM(In)=2n-1 if n≤33, as n → ∞ we have CM(In)∼2.089...n. The results have application to numerical integration. © 1992 Springer-Verlag New York Inc.