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A bound, and a conjecture, on the maximum lattice-packing density of a superball

Part of: Geometry of numbers

Published online by Cambridge University Press:  26 February 2010

J. A. Rush
J. A. Rush
Affiliation:
Department of Mathematics, GN-50, University of Washington, Seattle, WA 98195, U.S.A..
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Abstract

We obtain explicit lower bounds on the lattice packing densities δL of superballs G of quite a general nature, and we conjecture that as the dimension n approaches infinity, the bounds are asymptotically exact. If the conjecture were true, it would follow that the maximum lattice-packing density of the Iσ-ball is 2−n(1+σ(1)) for each σ in the interval 1 ≤ σ ≤ 2.

MSC classification

Secondary: 11H31: Lattice packing and covering
Type
Research Article
Information
Mathematika , Volume 40 , Issue 1 , June 1993 , pp. 137 - 143
Copyright
Copyright © University College London 1993

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References

1.Cassels, J. W. S.. An Introduction to the Geometry of Numbers (Springe-Verlag, N.Y., 1959).CrossRefGoogle Scholar
2.Conway, J. H. and Sloane, N. J. A.. Sphere Packings, Lattices and Groups (Springer-Verlag, N.Y., 1987).Google Scholar
3.Elkies, N. D., Odlyzko, A. M. and Rush, J. A.. On the packing densities of superballs and other bodies. Invent Math., 105 (1991), 613–639.CrossRefGoogle Scholar
4.Erdős, P., Gruber, P. M. and Hammer, J.. Lattice Points, Pitman Monograph 39 (Longman Scientific, with John Wiley & Sons, N.Y., 1989).Google Scholar
5.Gradshteyn, I. S. and Ryzhik, I. M.. Table of Integrals, Series and Products (Academic Press, New York, 1980), translated from Russian.Google Scholar
6.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers (Elsevier, North-Holland, Amsterdam, 1987).Google Scholar
7.Hlawka, E.. Zur Geometrie der Zahlen. Math. Zeitschr., 49 (1943), 285–312.CrossRefGoogle Scholar
8.Kabatiansky, G. A. and Levenshtein, V. I.. Bounds for packings on a sphere and in space (in Russian). Problemy Peredachy lnformatsii, 14 (1978), 3–25; English translation in Problems of Information Transmission, 14 (1978), 1–17.Google Scholar
9.Leech, J. and Sloane, N. J. A.. Sphere packing and error-correcting codes. Canad. J. Math., 23 (1971), 718–745.CrossRefGoogle Scholar
10.Minkowski, H.. Geometrie der Zahlen (B. G. Teubner, Leipzig, 1896).Google Scholar
11.Minkowski, H.. Gesammehe Abhandlungen (Chelsea, N.Y., reprint, 1969).Google Scholar
12.Rogers, C. A.. Packing and Covering (Cambridge University Press, 1964).Google Scholar
13.Rush, J. A. and Sloane, N. J. A.. An improvement to the Minkowski-Hlawka bound for packing superballs. Mathematika, 34 (1987), 8–18.CrossRefGoogle Scholar
14.Rush, J. A.. A lower bound on packing density. Invent. Math., 98 (1989), 499–509.CrossRefGoogle Scholar
15.Rush, J. A.. Thin lattice coverings. J. Land. Math. Soc, (2), 45 (1992), 193–200.CrossRefGoogle Scholar
16.Rush, J. A.. Constructive packings of cross polytopes. Mathematika, 38 (1991), 376–380.CrossRefGoogle Scholar