Arithmetic progressions and holomorphic phase retrieval

Abstract

We study the determination of a holomorphic function from its absolute value. Given a parameter $θ\in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $Λ\subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{ξz}, \, ξ\in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{|F(z)| : z \in e^{iθ}(\mathbb{R} + iΛ)\}$ if and only if $Λ$ 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.