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Lattice coverings of space

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
Affiliation:
University College, London, W.C.1.
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Let K be a n-dimensional convex body. A lattice Λ will be called a covering lattice for K (or simply a covering lattice), if each point of space has at least one representation in the form k + g where k ε K and g ε Λ. The density ϑ(K, Λ) of the resultant covering of the whole of space by the bodies of the form K + g with g ε Λ is, quite naturally, defined to be the ratio V(K)/d(Λ), where V(K) is the volume of K and d(Λ) is the determinant of Λ (i.e. the volume of its fundamental parallepipeds). Clearly this density is at least 1. A number of authors, in particular H. Davenport [3], R. P. Bambah and K. F. Roth [2], G. L. Watson [8], C. A. Rogers [4], [5] and W. Schmidt [7], have either constructed or proved the existence of lattices for which the density ϑ(K, Λ) is reasonably small. But, when n is large, all the densities obtained are of the form cn where c is a constant greater than 1. The object of this note is to obtain some rather stronger results. In the general case we prove the existence of covering lattices Λ with

Where

as n→∞.

Type
Research Article
Copyright
Copyright © University College London 1959

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References

1.Bambah, R. P. and Davenport, H., Journal London Math. Soc., 27 (1952), 224229.Google Scholar
2.Bambah, R. P. and Roth, K. F., Journal Indian Math. Soc., 16 (1952), 712.Google Scholar
3.Davenport, H., Rendiconti del C. Mat. di Palermo, Series 2, 1 (1952), 92107.Google Scholar
4.Rogers, C. A., Journal London Math. Soc., 33 (1958), 208212.Google Scholar
5.Rogers, C. A., Proc. London Math. Soc. (3), 8 (1958), 447465.Google Scholar
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7.Schmidt, W., “Masstheorie in der Geometrie der Zahlen”, to appear in Acta Math.Google Scholar
8.Watson, G. L., Rendiconti del C. Mat. di Palermo, Series 2, 5 (1956), 93100.Google Scholar