A Minkowski Theorem for Quasicrystals

The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {R}^n$$\end{document}. 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 quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.