A point set puzzle revisited

Two discrete point sets in R n are said to be homometric if their difference sets coincide. Homometric point sets were first studied in the 1930s in connection with the interpretation of x-ray diffraction patterns; today they appear in many contexts. Open questions still abound, even for point sets on the line. Under what conditions does a difference set S − S characterize S uniquely? If it does not, how can we find all the sets S i , i = 1 , … , that give rise to it, and how are these sets related?