Fourier quasicrystals and discreteness of the diffraction spectrum

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.