Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity

Abstract

We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure has pure point diffraction and satisfies the so called consistent phase property if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure has pure point diffraction and satisfies the consistent phase property independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias.