On higher dimensional arithmetic progressions in Meyer sets

TL;DR

This paper investigates the existence of higher dimensional arithmetic progressions in Meyer sets, establishing conditions under which such progressions with linearly independent ratios exist, and characterizing Meyer sets related to Euclidean model sets.

Contribution

It provides a characterization of Meyer sets containing higher dimensional arithmetic progressions with linearly independent ratios, linking their existence to the rank of the generated $bZ$-module.

Findings

Higher dimensional arithmetic progressions exist in Meyer sets if and only if the ratios are within the rank of the $bZ$-module.

The case of linearly dependent ratios over $bZ$ is trivial.

Characterization of Meyer sets as subsets of fully Euclidean model sets.

Abstract

In this paper we study the existence of higher dimensional arithmetic progression in Meyer sets. We show that the case when the ratios are linearly dependent over $\ZZ$ is trivial, and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda$ and a fully Euclidean model set $\oplam(W)$ with the property that finitely many translates of $\oplam(W)$ cover $\Lambda$, we prove that we can find higher dimensional arithmetic progressions of arbitrary length with $k$ linearly independent ratios in $\Lambda$ if and only if $k$ is at most the rank of the $\ZZ$-module generated by $\oplam(W)$. We use this result to characterize the Meyer sets which are subsets of fully Euclidean model sets.