On Pauli pairs and Fourier uniqueness problems

We investigate the concept of Pauli pairs and a discrete counterpart to it. In particular, we make substantial progress on the question of when a discrete Pauli pair is automatically a classical Pauli pair. Effectively, if one of the functions has space and frequency Gaussian decay, and one has that $|f| = |g|$ and $|\widehat{f}| = |\widehat{g}|$ on two sets which accumulate like suitable small multiples of $\sqrt{n}$ at infinity, then $|f| \equiv |g|$ and $|\widehat{f}| = |\widehat{g}|.$ Furthermore, we show that if one drops either the assumption that one of the functions has space-frequency decay or that the discrete sets accumulate at a high rate, then the desired property no longer holds. Our techniques are inspired by and directly connected to several recent results in the realm of Fourier uniqueness problems, and our results may be seen as a nonlinear generalization of those. As a consequence of said techniques, we are able to prove a sharp discrete version of Hardy's uncertainty principle.