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The Minkowski-Hlawka theorem

Published online by Cambridge University Press:  26 February 2010

C. A. Rogers
C. A. Rogers
Affiliation:
University College, London
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Minkowski's fundamental theorem and the Minkowski-Hlawka theorem play basic complementary roles in the Geometry of Numbers. Blichfeldt showed essentially that Minkowski's fundamental theorem was a simple consequence of a more general theorem, in which the convex body was replaced by any measurable set and the lattice was replaced by a discrete set of points having a definite asymptotic density. Hlawka himself showed that the Minkowski-Hlawka theorem could be proved in a slightly modified form, when the star body was replaced by any measurable set, but he did not replace the lattice by a more general set of points.

Type
Research Article
Information
Mathematika , Volume 1 , Issue 2 , June 1954 , pp. 111 - 124
Copyright
Copyright © University College London 1954

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References

page 111 note † See, for example, Hardy, G. H. and Wright, E. M., Theory of Numbers, 2nd Edit. (Oxford, 1945), Chapters 3 and 24.Google Scholar

page 111 note ‡ Hlawka, E., Math. Zeitschrift, 49 (1944), 285–312CrossRefGoogle Scholar; for a simple proof see, for example, Cassels, J. W. S., Proc. Cambridge Phil. Soc., 49 (1953), 165–166.CrossRefGoogle Scholar

page 111 note § Blichfeldt, H. F., Trans. American Math. Soc., 15 (1914), 227–235.CrossRefGoogle Scholar

page 111 note ‖ Loc. cit.

page 111 note ¶ Mahler, K., Journal London Math. Soc., 19 (1944), 201–205.CrossRefGoogle Scholar

page 111 note †† Weyl, H., in unpublished work; see Notes of the Seminar on Geometry of Numbers, The Institute for Advanced Study, Princeton, 1949, page 46.Google Scholar

page 111 note ‡‡ Rogers, C. A., see Notes of the Seminar on Geometry of Numbers, The Institute for Advanced Study, Princeton, 1949, pages 46–50.Google Scholar

page 111 note §§ Cassels, J. W. S., loc cit.Google Scholar

page 111 note ‖‖ There are many different ways of giving an equivalent definition of this upper density δ+(Λ): the same formula will, for example, serve to define δ+(Λ), if Σ(r) is used to denote any convex body containing a sphere of radius r.

page 112 note † Siegel, C. L., Annals of Math. (2), 46 (1945), 340–347.CrossRefGoogle Scholar

page 113 note † There are many equivalent definitions for the upper and lower directional densities. It may be noted that, if a set A is subjected to a translation, this does not change any directional density, and that, if two sets correspond to each other under a linear transformation of determinant 1, then their densities in corresponding directions are equal.