Deforming Meyer sets

TL;DR

This paper characterizes when linear deformations of Meyer sets remain Meyer sets and shows that pure point diffraction is preserved under injective deformations.

Contribution

It provides a necessary and sufficient condition for a linear deformation of a Meyer set to also be a Meyer set, extending understanding of their structural stability.

Findings

Deformations are Meyer sets if and only if certain conditions are met.

Injective deformations of Meyer sets preserve pure point diffraction.

The results clarify the stability of Meyer sets under linear transformations.

Abstract

A linear deformation of a Meyer set $M$ in $\RR^d$ is the image of $M$ under a group homomorphism of the group $[M]$ generated by $M$ into $\RR^d$. We provide a necessary and sufficient condition for such a deformation to be a Meyer set. In the case that the deformation is a Meyer set and the deformation is injective, the deformation is pure point diffractive if the orginal set $M$ is pure point diffractive.