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