The Structure of Homometric Sets

One of the fundamental problems of phase retrieval in spectroscopic analysis is of a combinatorial nature and can be solved using purely algebraic techniques. Given two sets A and B in some Euclidean space $R^n $, A and B are homometric if the sets of vector differences $\{ x - y:x,y \in A \}$ and $\{ x - y:x,y \in B \}$ are identical counting multiplicities. More generally, given two finite sums $A = \sum a_x \delta _x $ and $B = \sum a_x \delta _x $, where $a_x $, $b_x $ are integers and $\delta_x $ denotes the Dirac mass at $x \in R^n $, A and B are homometric if they have the same Patterson functions, i.e., for all $z \in R^n $, $\sum \{ a_x a_y :x - y = z \} = \sum \{ b_x b_y :x - y = z \}$. Using a variation on factorization of polynomials with integer coefficients, one can prove that A and B are homometric if and only if there exists two finite sums $C = \sum c_x \delta _x $ and $D = \sum d_x \delta _x $ such that A is the convolution $C * D$ and B is the convolution $C * D^* $, where $D^ * ( x ) =...