  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 solves what we call the phase problem if and only if it is
Besicovitch almost periodic. Finally, we show that a translation bounded
measure solves the phase problem 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.
