Fourier quasicrystals and discreteness of the diffraction spectrum

TL;DR

This paper proves that positive-definite measures with discrete support and spectrum can be expressed as finite combinations of Dirac combs, implying that certain quasicrystals with discrete diffraction spectra are necessarily periodic.

Contribution

It extends previous results by showing that measures with discrete support and spectrum are representable as finite Dirac comb combinations, revealing periodicity in some quasicrystals.

Findings

Measures with discrete support and spectrum are finite linear combinations of Dirac combs.

Hof's quasicrystals with discrete diffraction spectra are necessarily periodic.

The results generalize previous work by removing the assumption of spectrum discreteness.

Abstract

We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.