How Model Sets Can Be Determined by Their Two-point and Three-point Correlations

We show that real model sets with real internal spaces are determined, up to translation and changes of density 0, by their 2and 3-point correlations. We also show that there exist pairs of real (even 1D) aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose 2and 3-point correlations are identical but which are not related by either translation or inversion of their windows. All these examples are pure point diffractive. Placed in the context of ergodic uniformly discrete point processes, the result is that real point processes of model sets based on real internal windows are determined by their second and third moments.