On the orthogonality of measures of different spectral type with respect to twisted Eberlein convolution

In this paper we show that under suitable conditions on their Fourier--Bohr coefficients, the twisted Eberlein convolution of a measure with pure point diffraction spectra and a measure with continuous diffraction spectra is zero. In particular, the diffraction spectrum of a linear combinations of the two measures is simply the linear combinations of the two diffraction spectra with absolute value square coefficients.Comment: 24 pages, updated using the twisted version of the Eberlein convolutio

Why do Meyer sets diffract?

Given a weak model set \oplam(W) in a locally compact Abelian group we construct a relatively dense set of common Bragg peaks for all subsets \Lambda \subseteq \oplam(W) with non-trivial Bragg spectrum. Next, for each \eps>0 we construct a relatively dense set P_\eps which are common \eps-norm almost periods for the diffraction, pure point, absolutely continuous and singular continuous spectrum, respectively, for all subsets \Lambda \subseteq \oplam(W). We use the Fibonacci model set to illustrate these phenomena, and extend the results to arbitrary weighted Dirac combs with weak Meyer set support. We complete the paper by discussing some extensions of the existence of the generalized Eberlein decomposition for measures with weak Meyer set support.Comment: 34 pages, 1 figur