A Minkowski theorem for Meyer sets

Abstract

In this paper, we generalize Minkowski's theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in $\mathbb{R}^n$. In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for Meyer sets, which bounds from below the frequency of differences appearing in the Meyer set and belonging to a centrally symmetric convex body. In the later part of the paper, we develop quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.