Lattice Points in Large Borel Sets and Successive Minima

Let B be a Borel set in Ed with volume V (B) = . It is shown that almost all lattices L in Ed contain infinitely many pairwise disjoint d-tuples, that is sets of d linearly independent points in B. A consequence of this result is the following: let S be a star body in Ed with V (S) = . Then for almost all lattices L in Ed the successive minima 1(S, L), . . . , d(S, L) of S with respect to L are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the semi-continuity of the successive minima. Key words. Borel sets, star bodies, lattices, successive minima, measure, Baire category, semicontinuity. MSC 2000. 11H16, 11H50, 11J25.

• Type: Journal
• Year: 2006
• Where: DCG
• Authors: Iskander Aliev, Peter M. Gruber
• Keywords: DCG 2006, Linearly Independent, Star Body, Successive Minima