Pure Point Diffractive Substitution Delone Sets Have the Meyer Property

We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J.C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra. The first author acknowledges support from the NSERC post-doctoral fellowship and thanks the University of Washington and the University of Victoria for being the host universities of the fellowship. The second author is grateful to the Weizmann Institute of Science where he was a Rosi and Max Varon Visiting Professor when this work was completed. He was also supported in part by NSF Grant DMS 0355187.