Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity

TL;DR

This paper establishes deep connections between pure point diffraction and various notions of almost periodicity (mean, Besicovitch, Weyl) for translation bounded measures, resolving key theoretical questions in diffraction theory.

Contribution

It characterizes pure point diffraction and the phase problem in terms of different types of almost periodicity, providing new theoretical insights and solutions to longstanding questions.

Findings

Pure point diffraction iff measure is mean almost periodic

Phase problem solved iff measure is Besicovitch almost periodic

Phase problem independent of van Hove sequence iff measure is Weyl almost periodic

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 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.