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.

References in corpus (8)

• Pure Point Dynamical and Diffraction Spectra
• Dynamics on the graph of the torus parametrisation
• Dense Dirac combs in Euclidean space with pure point diffraction
• On pattern entropy of weak model sets
• On the Fourier Analysis of Measures with Meyer Set Support
• A short guide to pure point diffraction in cut-and-project sets
• Equicontinuous delone dynamical systems
• Diffraction and Palm measure of point processes

Cited by in corpus (10)

• On norm almost periodic measures
• Leptin densities in amenable groups
• Multi-dimensional almost periodic type functions and applications
• On arithmetic progressions in non-periodic self-affine tilings
• On arithmetic progressions in model sets
• Spectrum of weak model sets with Borel windows
• Model sets with precompact Borel windows
• Why do Meyer sets diffract?
• Inter model sets in ${\mathbb{R}}^d$ are model sets
• Semi-measures and their Fourier transform