A point set puzzle revisited

Two discrete point sets in Rⁿ 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?