Equivalence between pure point diffractive sets and cut-and-project sets on substitution tilings

Quasicrystals are characterized by the property of pure point diffractive spectrum mathematically. We look at substitution tilings and characterize the pure point diffractive spectrum by regular model sets defined from a cut-and-project scheme. The cut-and-project scheme is built with a physical space R-d and an internal space which is a product of a Euclidean space and a profinite group. The assumptions we make here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with same multiplicity. We give a precise argument for the proof on a specific example.