Eberlein decomposition for PV inflation systems

The Dirac combs of primitive Pisot--Vijayaraghavan (PV) inflations on the real line or, more generally, in $\mathbb{R}^d$ are analysed. We construct a mean-orthogonal splitting for such Dirac combs that leads to the classic Eberlein decomposition on the level of the pair correlation measures, and thus to the separation of pure point versus continuous spectral components in the corresponding diffraction measures. This is illustrated with two guiding examples, and an extension to more general systems with randomness is outlined.