On Weakly Almost Periodic Measures

TL;DR

This paper investigates the properties of weakly almost periodic measures, revealing their dynamical and spectral characteristics, and establishing their stability and convolution properties, with implications for diffraction and aperiodic order.

Contribution

It provides a comprehensive analysis of weakly almost periodic measures, including their dynamical hulls, spectral properties, and convolution behavior, extending understanding in aperiodic order and diffraction theory.

Findings

The dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system.

The hull is minimal if and only if the measure is strongly almost periodic.

The Eberlein convolution of two weakly almost periodic measures is unique and strongly almost periodic.

Abstract

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point respectively continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.