Arithmetic progressions and holomorphic phase retrieval

We study the determination of a holomorphic function from its absolute value. Given a parameter $\theta \in R$, we derive the following characterization of uniqueness in terms of rigidity of a set $\Lambda \subseteq R$: if $F$ is a vector space of entire functions containing all exponentials $e^{\xi z},\xi \in C\setminus \left\{0\right\}$, then every $F\in F$ is uniquely determined up to a unimodular phase factor by $\left\{\right|F\left(z\right)|:z\in e^{i\theta }(R+i\Lambda )\}$ if and only if $\Lambda$ is not contained in an arithmetic progression $aZ+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli-type uniqueness problems. For instance, $Z\times \stackrel{\sim }{Z}$ is a uniqueness set for the Gabor phase retrieval problem in $L^2\left(R_{+}\right)$, provided that $\stackrel{\sim }{Z}$ is a suitable perturbation of the integers.