Model sets, almost periodic patterns, uniform density and linear maps

TL;DR

This paper explores the geometric properties of model sets and almost periodic patterns, demonstrating their uniform density and stability under linear map discretizations, with implications for understanding their structure and transformations.

Contribution

It establishes that model sets are almost periodic patterns with uniform density and proves their stability under discretizations of linear maps.

Findings

Model sets are almost periodic patterns with uniform density.

Almost periodic patterns are stable under linear map discretizations.

The geometric properties of model sets are characterized under minimal hypotheses.

Abstract

This article consists in two independent parts. In the first one, we investigate the geometric properties of almost periodicity of model sets (or cut-and-project sets, defined under the weakest hypotheses); in particular we show that they are almost periodic patterns and thus possess a uniform density. In the second part, we prove that the class of model sets and almost periodic patterns are stable under discretizations of linear maps.