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

Fourier quasicrystals and discreteness of the diffraction spectrum