Deforming Meyer sets

A linear deformation of a Meyer set $M$ in ${\text{RR}}^d$ is the image of $M$ under a group homomorphism of the group $[M]$ generated by $M$ into ${\text{RR}}^d$. 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.