On the Covering Multiplicity of Lattices

Let the lattice Λ have covering radius R, so that closed balls of radius R around the lattice points justcover the space. The covering multiplicity CM(Λ) is the maximal number of times the interiors of theseballs overlap. We show that the least possible covering multiplicity for an n-dimensional lattice is n ifn ≤ 8, and conjecture that it exceeds n in all other cases. We determine the covering multiplicity of theLeech lattice and of the lattices I n , A n , D n , E n and their duals for small values of n. Although it appearsthat CM(I n ) = 2 − 1 if n ≤ 33, as n → ∞ we have CM(I n ) ∼ 2. 089 . . . . The results have application tonumerical integration. ____________* This paper appeared in Discrete and Computational Geometry, vol. 8 (1992), pp. 109-130.