On the Bragg Diffraction Spectra of a Meyer Set

TL;DR

This paper studies the diffraction spectra of Meyer sets, revealing that subsets of Bragg peaks above certain intensities are also Meyer sets and that small modifications preserve the dense Bragg peak structure.

Contribution

It demonstrates that high-intensity Bragg peaks form Meyer sets and that minor modifications to Meyer sets retain their diffraction properties.

Findings

High-intensity Bragg peaks form Meyer sets.

Small modifications do not destroy the dense Bragg spectrum.

Results deepen understanding of aperiodic crystal diffraction.

Abstract

Meyer sets have a relatively dense set of Bragg peaks and for this reason they may be considered as basic mathematical examples of (aperiodic) crystals. In this paper we investigate the pure point part of the diffraction of Meyer sets in more detail. The results are of two kinds. First we show that given a Meyer set and any intensity a less than the maximum intensity of its Bragg peaks, the set of Bragg peaks whose intensity exceeds a is itself a Meyer set (in the Fourier space). Second we show that if a Meyer set is modified by addition and removal of points in such a way that its density is not altered too much (the allowable amount being given explicitly as a proportion of the original density) then the newly obtained set still has a relatively dense set of Bragg peaks.