Wigner transform and quasicrystals (Q2067066)

The authors show that if the Wigner transform of a tempered distribution \(\mu\in \mathcal{S}'(\mathbb{R}^{d})\) is a measure supported in the Cartesian product of two uniformly distributed subsets \(A\subset\mathbb{R}^{d}\) and \(B\subset\mathbb{R}^{d}\), then \(\mu\) and its Fourier transform are measures that satisfy \(\operatorname{supp} \mu\subset A\) and \(\operatorname{supp} \hat{\mu}\subset B\). Hence, the measure \(\mu\) is a Fourier quasicrystal. In dimension \(d=1\), they also give a version of this result for matrix-Wigner transforms.