Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets

The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.