Phase retrieval from short-time Fourier transform in LCA groups

We study the short-time Fourier transform phase retrieval problem in locally compact abelian groups. Using probabilistic methods, we show that for a large class of groups $G$ and compact subsets $K\subseteq G$ there exists a window function and a uniformly discrete set in $G\times \widehat{G}$ allowing phase retrieval in $L^2(K)$. We also study the obstructions for STFT phase retrieval on $L^2(G)$, motivating the restriction to compactly supported function spaces.