On non-uniqueness of phase retrieval for discrete and continuous Fourier transforms

We consider the problem of phase retrieval for continuous and discrete Fourier transforms in multiple dimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work by Novikov and Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. We also consider the problem of phase retrieval with background information. We show that this problem is not uniquely solvable if the known and unknown parts of the function are not sufficiently disjoint.