On embedding of repetitive Meyer multiple sets into model multiple sets

TL;DR

This paper establishes that for repetitive Meyer multiple sets in Euclidean space, being a model multiple set is equivalent to the associated dynamical system being almost automorphic, achieved through embedding into a smaller eigenvalue group model.

Contribution

It proves the equivalence between being a model multiple set and the almost automorphic property of the dynamical system for repetitive Meyer multiple sets.

Findings

Repetitive Meyer multiple sets can be embedded into model multiple sets.

Embedding reduces the group of topological eigenvalues.

Equivalence between model multiple sets and almost automorphy is established.

Abstract

Model sets are always Meyer sets but the converse is generally not true. In this work we show that for a repetitive Meyer multiple sets of $\mathbb{R}^d$ with associated dynamical system $(\mathbb{X}, \mathbb{R}^d)$, the property of being a model multiple set is equivalent for $(\mathbb{X}, \mathbb{R}^d)$ to be almost automorphic. We deduce this by showing that a repetitive Meyer multiple set can always be embedded into a repetitive model multiple set having a smaller group of topological eigenvalues.