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 2- and 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 2- and 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.