Complete subsets of a diffraction pattern

A positive atomic density rho (r) = Sigma (N)(j=1)n(j)delta (r - r(j)) in a D-dimensional space can be exactly reconstructed from an appropriate finit e subset (complete set) of its Fourier series coefficients {U-h}(h is an el ement of ZD) or even (limited to the support {r(j)}(j=1,..., N)) from a fin ite subset of moduli \U-h\. It is necessary first to determine a complete s et of Fourier coefficients U-h (possibly inside the unavoidable high-resolu tion cut-off parallel toh parallel to < L) and then, by these coefficients, to determine the unknown density. We report some procedures which are able to single out complete sets. They are based on a property of Goedkoop's ve ctor lattice {\A(h))}(h<is an element of>ZD), defined so that (A(h)/A(k)) = Uk-h. The property states that if the vector with index h* = (h(1)*,..., h (D)*) is a linear combination of the vectors relevant to a set of indices w ith a particular shape, then all the vectors relevant to the hyperquadrant Q(h)* = {h\h(alpha) greater than or equal to h(a)*, alpha = 1,..., D) are l inear combinations of the vectors relevant to Q(o)\Q(h)*. Moreover, the det ermination of rho from a complete set passes through the solution of a syst em of polynomial equations in D variables, whose roots determine the positi on vectors. We show how to convert this problem into the simpler problem of sequentially solving a set of polynomial equations in one variable.