Pure point diffraction and almost periodicity

Abstract: This article deals with pure point diffraction and its connection to various notions of almost periodicity. We explain why the Fibonacci chain does not fit into the classical class of Bohr almost periodicity and how it fits into the classes of mean, Besicovitch and Weyl almost periodic point sets. We report on recent results which characterize pure point diffraction as mean almost periodicity of the underlying structure, and discuss how the complex amplitudes fit into this picture.