Arithmetic progressions and holomorphic phase retrieval

We study the determination of a holomorphic function from its absolute value. Given a parameter , we derive the following characterization of uniqueness in terms of rigidity of a set : if is a vector space of entire functions containing all exponentials , then every is uniquely determined up to a unimodular phase factor by if and only if is not contained in an arithmetic progression . Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli‐type uniqueness problems. For instance, is a uniqueness set for the Gabor phase retrieval problem in , provided that is a suitable perturbation of the integers.