  We consider the finite alphabet phase retrieval problem: recovering a signal
whose entries lie in a small alphabet of possible values from its Fourier
magnitudes. This problem arises in the celebrated technology of X-ray
crystallography to determine the atomic structure of biological molecules. Our
main result states that for generic values of the alphabet, two signals have
the same Fourier magnitudes if and only if several partitions have the same
difference sets. Thus, the finite alphabet phase retrieval problem reduces to
the combinatorial problem of determining a signal from those difference sets.
Notably, this result holds true when one of the letters of the alphabet is
zero, namely, for sparse signals with finite alphabet, which is the situation
in X-ray crystallography.
