Fourier Uniqueness and Nonuniqueness Pairs

Motivated by a discovery by Radchenko and Viazovska and by a work by Ramos and Sousa, we find conditions sufficient 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 theorem can be upgraded to the frame bound and an interpolation formula, which in turn produce an abundance of Poisson-like formulas (a.k.a. "crystalline measures"). The talk is based on a joint work with A. Kulikov and F. Nazarov https://arxiv.org/abs/2306.14013 .