# 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.

## Key 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.