Meyer sets, topological eigenvalues, and Cantor fiber bundles

TL;DR

This paper characterizes Meyer sets using topological eigenvalues and fiber bundle structures, revealing new insights into their topological and spectral properties, and providing counterexamples to previous assumptions.

Contribution

It introduces two new characterizations of Meyer sets based on topological conjugacy and eigenvalues, and presents counterexamples clarifying their spectral distinctions.

Findings

Repetitive Delone sets with $d$ linearly independent topological eigenvalues are conjugate to Meyer sets.

Existence of sets topologically conjugate to Meyer sets but not Meyer themselves.

A diffractive set that is not Meyer, answering Lagarias' question.

Abstract

We introduce two new characterizations of Meyer sets. A repetitive Delone set in $\R^d$ with finite local complexity is topologically conjugate to a Meyer set if and only if it has $d$ linearly independent topological eigenvalues, which is if and only if it is topologically conjugate to a bundle over a $d$-torus with totally disconnected compact fiber and expansive canonical action. "Conjugate to" is a non-trivial condition, as we show that there exist sets that are topologically conjugate to Meyer sets but are not themselves Meyer. We also exhibit a diffractive set that is not Meyer, answering in the negative a question posed by Lagarias, and exhibit a Meyer set for which the measurable and topological eigenvalues are different.