Phaseless sampling on square-root lattices

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $g \in L^2(\mathbb{R}^d)$ and which sampling sets $\Lambda \subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$|V_gf(\Lambda)| = \left \{ |V_gf(\lambda)| : \lambda \in \Lambda \right \},$$ where $V_gf$ denotes the short-time Fourier transform (STFT) of $f$ with respect to $g$. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $\Lambda$ is a lattice, i.e $\Lambda = A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$\Lambda = A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \},$$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians.