Quasicrystals and model sets on substitution point sets

The typical property of a quasicrystal is that it lacks periodic order but its diffraction pattern has a strong component of Bragg peaks. We consider the diffraction pattern consisting purely of Bragg peaks and call it pure point diffraction. It has been known that any regular model sets are pure point diffractive. But the converse has been a conjecture. Here we show that in substitution point sets, the sets are pure point diffractive if and only if they are model sets.