Fourier quasicrystals with unit masses

Measures with discrete support and spectrum which generate classical Poisson formulas were studied by J.-P. Kahane, S. Mandelbrojt, A. Guinand in the 50-ies and later by Y. Meyer. The revival of interest in such measures is due to the discovery of physical quasicrystals by D. Shechtman in the 80-ies. The important case where both the support and spectrum of the measure are locally finite sets has been actively studied in the last years. A recent progress is due to P. Kurasov and P. Sarnak who constructed examples of aperiodic measures on the real axis with unit masses and uniformly discrete support. I will discuss a joint result with A. Olevskii which characterizes all such measures ( https://arxiv.org/abs/2006.12037 <https://arxiv.org/abs/2006.12037>, https://arxiv.org/abs/2009.12810 <https://arxiv.org/abs/2009.12810> )
(The talk is based on joint work with Alexander Olevskii)