Note on the set of Bragg peaks with high intensity

TL;DR

This paper investigates the structure of high-intensity Bragg peaks in diffraction patterns of Delone sets, establishing that dense high-intensity peaks form Meyer sets and offering a new characterization of Meyer sets via measures.

Contribution

It introduces a novel link between high-intensity Bragg peaks and Meyer sets, providing a new measure-based characterization of Meyer sets in diffraction analysis.

Findings

High-intensity Bragg peaks form Meyer sets if relatively dense.

Provides a new measure-theoretic characterization of Meyer sets.

Studies positive definite measures relevant to diffraction patterns.

Abstract

We consider diffraction of Delone sets in Euclidean space. We show that the set of Bragg peaks with high intensity is always Meyer (if it is relatively dense). We use this to provide a new characterization for Meyer sets in terms of positive and positive definite measures. Our results are based on a careful study of positive definite measures, which may be of interest in its own right.