  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.
