Lattice Points in Large Borel Sets and Successive Minima

Abstract

Let $B$ be a Borel set in ${\mathbb{E}}^d$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in ${\mathbb{E}}^d$ 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 ${\mathbb{E}}^d$ with $V(S)=\infty$. Then for almost all lattices $L$ in ${\mathbb{E}}^d$ the successive minima ${\lambda }_1(S,L),...,{\lambda }_d(S,L)$ 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.

8 pages