Square-root lattices: where do they appear?

Question

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is known as phase retrieval. Today this article was brought to my attention: Phaseless sampling on square-root lattices.

In this article the authors show a theorem that I find very interesting: they show that the absolute value of the windowed Fourier transform $$ V_gf(x,\omega) = \int f(t)\overline{g(t-x)} e^{-2\pi i \omega t} \, dt, \ \ \ \ \ \ \ \ f,g \in L^2(\mathbb R^n) $$ can be sampled on a square-root lattice $A \sqrt{\mathbb Z}^{2n}$ and from these samples any function that is in $L^2(\mathbb R^n)$ can be recovered uniquely (except for a phase factor). This is the first time that I witnessed a result and a sampling method of this kind. Is any of you aware of related results? Do square-root lattices appear somewhere else in mathematics or research on phase retrieval?

Thank you!

Answer 1

Lattices made from the square roots of elements in a given lattice can be used to identify unique factorizations in some quadratic integer fields where a unique factorization otherwise does not exist. For instance, the number $66$ has nonunique factorization among algebraic integers having the form $a+b\sqrt{-30}$, but by bringing in appropriate square-root lattices we can make one factorization from which all the nonunique factorizations in the original $\mathbb{Z}[\sqrt{-30}]$ lattice can be derived and related to each other. See https://math.stackexchange.com/a/4439849/248217 for details.