This is a report on a joint work in progress with Aleksei Kulikov and Fedor Nazarov.

Motivated by recent works by Radchenko and Viazovska and by Ramos and Sousa, we find sufficient conditions for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other.

The uniqueness can be upgraded to an interpolation formula, which in turn produces an abundance of "crystalline measures", i.e., discrete measures, which are tempered distribution, whose Fourier transform is also a discrete measure.