Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions

It was recently shown that functions in $L^4([-B,B])$ can be uniquely recovered up to a global phase factor from the absolute values of their Gabor transforms sampled on a rectangular lattice. We prove that this remains true if one replaces $L^4([-B,B])$ by $L^p([-B,B])$ with $p \in [1,\infty]$. To do so, we adapt the original proof by Grohs and Liehr and use a classical sampling result due to Beurling. Furthermore, we present a minor modification of a result of M\"untz-Sz\'asz type by Zalik. Finally, we consider the implications of our results for more general function spaces obtained by applying the fractional Fourier transform to $L^p([-B,B])$ and for more general nonuniform sampling sets.