Pure point diffraction and Poisson summation

TL;DR

This paper establishes a fundamental link between diffraction patterns of regular model sets and the Poisson Summation Formula, extending results to some non-regular sets using advanced Fourier analysis techniques.

Contribution

It proves the equivalence of diffraction formulas and Poisson Summation for regular model sets and extends diffraction results to certain non-regular weak model sets.

Findings

Diffraction formula for regular model sets is equivalent to Poisson Summation.

Extended diffraction results to some non-regular weak model sets.

Utilized Fourier analysis of unbounded measures on locally compact abelian groups.

Abstract

We prove that the diffraction formula for regular model sets is equivalent to the Poisson Summation Formula for the underlying lattice. This is achieved using Fourier analysis of unbounded measures on locally compact abelian groups as developed by Argabright and de Lamadrid. We also obtain diffraction results for certain classes of non-regular so-called weak model sets.