  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.
