On Weighted Dirac Combs Supported Inside Model Sets

Abstract: In this paper we prove that given a weakly almost periodic measure $\mu$ supported inside some model set $\Lambda(W)$ with closed window $W$, then the strongly almost periodic component $\mu_S$ and the null weakly almost periodic component $\mu_0$ are both supported inside $\Lambda(W)$. As a consequence we prove that given any translation bounded measure $\omega$, supported inside some model set, then each of the pure point diffraction spectrum $\hat{\gamma}_{pp}$ and the continuous diffraction spectrum $\hat{\gamma}_c$ is either trivial or have a relatively dense support.