Arithmetic progressions and holomorphic phase retrieval.

We study the determination of a holomorphic function from its absolute value. Given a parameter θ∈R$\theta \in \mathbb {R}$, we derive the following characterization of uniqueness in terms of rigidity of a set Λ⊆R$\Lambda \subseteq \mathbb {R}$: if F$\mathcal {F}$ is a vector space of entire functions containing all exponentials eξz,ξ∈C∖{0}$e^{\xi z}, \, \xi \in \mathbb {C} \setminus \lbrace 0 \rbrace$, then every F∈F$F \in \mathcal {F}$ is uniquely determined up to a unimodular phase factor by {|F(z)|:z∈eiθ(R+iΛ)}$\lbrace |F(z)|: z \in e^{i\theta }({\mathbb {R}}+ i \Lambda) \rbrace$ if and only if Λ$\Lambda$ is not contained in an arithmetic progression aZ+b$a\mathbb {Z}+b$. Leveraging this insight, we establish a series of consequences for Gabor phase retrieval and Pauli‐type uniqueness problems. For instance, Z×Z∼${\mathbb {Z}}\times \tilde{{\mathbb {Z}}}$ is a uniqueness set for the Gabor phase retrieval problem in L2(R+)$L^2({\mathbb {R}}_+)$, provided that Z∼$\tilde{{\mathbb {Z}}}$ is a suitable perturbation of the integers. [ABSTRACT FROM AUTHOR]