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 λ1(S,L),..., λd(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.