How model sets can be determined by their two-point and three-point correlations

TL;DR

This paper demonstrates that real model sets with real internal spaces are uniquely determined by their two- and three-point correlations, but counterexamples exist when internal spaces include finite cyclic groups.

Contribution

It establishes that real model sets are determined by their second and third correlations, and provides counterexamples involving internal spaces with cyclic groups.

Findings

Real model sets with real internal spaces are determined by two- and three-point correlations.

Counterexamples show non-uniqueness when internal spaces include cyclic groups.

All examples discussed are pure point diffractive.

Abstract

We show that real model sets with real internal spaces are determined, up to translation and changes of density zero by their two- and three-point correlations. We also show that there exist pairs of real (even one dimensional) aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose two- and three-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.