Pure point/Continuous decomposition of translation-bounded measures and diffraction

Abstract

In this work we consider translation-bounded measures over a locally compact Abelian group $\mathbb{G}$, with particular interest for their so-called diffraction. Given such a measure $\Lambda$, its diffraction $\widehat{\gamma}$ is another measure on the Pontryagin dual $\widehat{\mathbb{G}}$, whose decomposition into the sum $\widehat{\gamma} = \widehat{\gamma}_{\mathrm{p}}+ \widehat{\gamma}_{\mathrm{c}}$ of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of $\widehat{\gamma}$ lifts to $\Lambda$ itself, that is to say, whether there exists a decomposition $\Lambda = \Lambda _{\mathrm{p}} + \Lambda _{\mathrm{c}}$, where $\Lambda _{\mathrm{p}}$ and $\Lambda _{\mathrm{c}}$ are translation-bounded measures having diffraction $\widehat{\gamma}_{\mathrm{p}}$ and $\widehat{\gamma}_{\mathrm{c}}$ respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.