Arithmetic progressions and holomorphic phase retrieval

Abstract

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.

14 pages