Lattice Points in Large Borel Sets and Successive Minima

Details Lattice Points in Large Borel Sets and Successive Minima Lattice Points in Large Borel Sets and Successive Minima

2005-10-08

arxiv.org/abs/math/0510163v1

Mathematics

Number Theory

8 pages

Scientific paper

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.

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