  Sampled Gabor phase retrieval - the problem of recovering a square-integrable
signal from the magnitude of its Gabor transform sampled on a lattice - is a
fundamental problem in signal processing, with important applications in areas
such as imaging and audio processing. Recently, a classification of
square-integrable signals which are not phase retrievable from Gabor
measurements on parallel lines has been presented. This classification was used
to exhibit a family of counterexamples to uniqueness in sampled Gabor phase
retrieval. Here, we show that the set of counterexamples to uniqueness in
sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal
to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to
a better understanding of the fundamental limits of sampled Gabor phase
retrieval.
