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On The Critical Lattices of Arbitrary Point Sets

Published online by Cambridge University Press:  20 November 2018

K. Mahler
K. Mahler*
Affiliation:
University of Manchester
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In this note, I shall establish necessary and sufficient conditions for the existence of critical lattices of an arbitrary point set, and I shall construct a non-trivial example of a point set without any critical lattice. In a previous paper, I proved that every star body of the finite type possesses at least one critical lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 78 - 87
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 4'On lattice points in w-dimensional star bodies, I,” Proc. Royal Soc, A, 187 (1946), 151-187. The letters LP will be used to mark references to this paper.

2 A sequence of lattices A1, A2, A3, … is said to be bounded if (i) the determinants d(Λr) are bounded, and (ii) no point P ≠ 0 of these lattices lies in a certain neighbourhood of 0.(LP, Definition 1, p. 155.)

3 It is possible to select from any bounded sequence of lattices a subsequence tending to a limiting lattice. (LP, Theorem 2, p. 156.)

4 LP, Theorem 8, p. 159.

5 The conditions (I) and (II) are satisfied if, e.g.

(r = 1, 2, 3,…), as is trivial for (I), and follows for (II) from the transcendency of e.

6 Geometrie der Zahlen,§ 46.