A Minkowski theorem for Meyer sets

TL;DR

This paper extends Minkowski's theorem to Meyer sets, providing bounds on difference frequencies within convex bodies and applying these results to Diophantine approximation and linear map discretization.

Contribution

It introduces a Minkowski theorem for Meyer sets, replacing lattices with more general sets that have a density concept, and explores applications in number theory and linear algebra.

Findings

Bounds on difference frequencies in Meyer sets within convex bodies

Applications to Diophantine approximation

Discretization of linear maps

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.