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On Weighted Dirac Combs Supported Inside Model Sets

In this paper we prove that given a weakly almost periodic measure $μ$ supported inside some model set $Λ(W)$ with closed window $W$, then the strongly almost periodic component $μ_S$ and the null weakly almost periodic component $μ_0$ are both supported inside $Λ(W)$. As a consequence we prove that given any translation bounded measure $ω$, supported inside some model set, then each of the pure point diffraction spectrum $\hatγ_{pp}$ and the continuous diffraction spectrum $\hatγ_c$ is either trivial or have a relatively dense support.