On the covering multiplicity of lattices |
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Authors: | J. H. Conway N. J. A. Sloane |
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Affiliation: | (1) Mathematics Department, Princeton University, 08540 Princeton, NJ, USA;(2) Mathematical Sciences Research Center, AT&T Bell Laboratories, 07974 Murray Hill, NJ, USA |
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Abstract: | Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI n, An, Dn, En and their duals for small values ofn. Although it appears thatCM(I n)=2 n−1 ifn≤33, asn → ∞ we haveCM(I n)∼2.089... n . The results have application to numerical integration. |
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