A point set puzzle revisited

Two discrete point sets in RnRn 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−SS−S characterize SS uniquely? If it does not, how can we find all the sets Si,i=1,…Si,i=1,…, that give rise to it, and how are these sets related?