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Which Meyer sets are regular model sets? A characterization via almost periodicity

Daniel Lenz, Christoph Richard, Nicolae Strungaru

Abstract

In 2012, Meyer introduced the notions of generalized almost periodic measure and almost periodic pattern and proved that regular model sets in Euclidean space are almost periodic patterns. Here, we prove the converse in a slightly more general setting. Specifically, we show that a Meyer set in any $σ$-compact locally compact abelian group is a regular model set if and only if it is an almost periodic pattern.

Which Meyer sets are regular model sets? A characterization via almost periodicity

Abstract

In 2012, Meyer introduced the notions of generalized almost periodic measure and almost periodic pattern and proved that regular model sets in Euclidean space are almost periodic patterns. Here, we prove the converse in a slightly more general setting. Specifically, we show that a Meyer set in any -compact locally compact abelian group is a regular model set if and only if it is an almost periodic pattern.

Paper Structure

This paper contains 15 sections, 23 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Setting and key players
  3. Functions, measures, and sets
  4. Almost periodic measures and their mean
  5. Meyer sets, cut-and-project schemes, and the map $\varOmega$
  6. The core of the argument
  7. G-a-p measures supported within Meyer sets
  8. Examples and counter-examples
  9. Meyer sets in a lattice
  10. On regular windows
  11. Regular inter model sets
  12. G-a-p measures and Weyl almost periodicity
  13. More about $\hbox{universal}$ cut-and-project schemes
  14. Uniqueness and factorization
  15. Characterizations

Key Result

Lemma 2.1

There exists a unique linear map $M : \mathcal{SAP}(G)\longrightarrow {\mathbb C}$ such that for all $\mu \in \mathcal{SAP}(G)$ and $\varphi \in C_{\mathsf{c}}(G)$. The map $M$ satisfies the following properties:

Theorems & Definitions (61)

  • Lemma 2.1: The mean on $\mathcal{SAP} (G)$
  • proof
  • Remark 2.2
  • Proposition 2.3: Computing the mean via averaging
  • Definition 2.4: Generalized almost periodicity and almost periodic patterns
  • Remark 2.5
  • Definition 2.6: Meyer set
  • Remark 2.7
  • Definition 2.8: Cut-and-project scheme over $G$
  • Definition 2.9: Regular model set
  • ...and 51 more