A short guide to pure point diffraction in cut-and-project sets

TL;DR

This paper provides an elementary proof of the diffraction formula for regular cut-and-project sets, clarifying the relationship between quasicrystal diffraction and their underlying lattices, with illustrative examples.

Contribution

It offers a new, elementary proof of the diffraction formula for regular cut-and-project sets using Fourier analysis, enhancing understanding of quasicrystal diffraction.

Findings

Elementary proof of diffraction formula based on Bochner's theorem

Clarification of the link between quasicrystal diffraction and lattice diffraction

Explicit examples illustrating the diffraction properties

Abstract

We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochner's theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples.