Dense Dirac combs in Euclidean space with pure point diffraction

TL;DR

This paper extends the mathematical diffraction theory of regular model sets to dense point sets in Euclidean space, providing insights into quasicrystallographic structures and their diffraction spectra.

Contribution

It introduces a new theoretical result that generalizes pure point diffraction to dense Euclidean point sets, inspired by quasicrystallographic random tilings.

Findings

Pure point diffraction spectrum for dense Euclidean point sets

Extension of regular model set results to more general dense sets

Relevance to quasicrystallographic random tilings

Abstract

Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact Abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean space, which is motivated by the study of quasicrystallographic random tilings.