Why do Meyer sets diffract?

TL;DR

This paper investigates the diffraction properties of Meyer sets, constructing common Bragg peaks and almost periods for their subsets, and extends these results to weighted Dirac combs, enhancing understanding of their spectral characteristics.

Contribution

It introduces a framework for analyzing diffraction spectra of Meyer sets and their subsets, including the construction of common Bragg peaks and almost periods, and extends results to weighted Dirac combs.

Findings

Existence of common Bragg peaks for all subsets of a Meyer set.

Construction of common norm almost periods for various spectral components.

Extension of results to weighted Dirac combs supported on Meyer sets.

Abstract

Given a weak model set in a locally compact Abelian, group we construct a relatively dense set of common Bragg peaks for all its subsets that have non-trivial Bragg spectrum. Next, we construct a relatively dense set of common norm almost periods for the diffraction, pure point, absolutely continuous and singular continuous spectrum, respectively, of all its subsets. We use the Fibonacci model set to illustrate these phenomena. We extend all these results to arbitrary translation bounded weighted Dirac combs supported within some Meyer set. We complete the paper by discussing extensions of the existence of the generalized Eberlein decomposition for measures supported within some Meyer set.