Consider the problem of recovering a set of real numbers X from the knowledge of its unlabeled set of differences xi --q > xi , xj E x, 1 <z.,j<N.
(1)
This problem comes up in different setups, among them in the so-called "phase problem in crystallography"; see [3] and the references given there. In that case one attempts to determine X, up to a congruency, from the modulus of the "amplitude function" h real.
The square of this modulus is the "intensity" and it is plain that the knowledge of I(h), f or enough values of A, is equivalent to that of the unlabeled differences (1).
One knows that the problem does not have a unique solution. There is a systematic way of producing lots of sets which are not congruent but have the same set of differences; see [2]. Quite recently, Bloom [I] found a new pair of such sets which adds new interest to the field. Indeed Piccard [4, p. 311 presents a result to the effect that if the nonzero elements in (1) are all different then one can reconstruct X from (1). The pair found by Bloom shows that Piccard's result does not hold.
It is not hard to find conditions, like demanding that the Xj , 1 < j < N, be linearly independent over the integers, that guaranteed that X can be found from (1). But all of these conditions are too strong, and a "good" sufficient condition is still lacking, This is particularly true when the xj's are taken to be integers, an assumption we make from now on.