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In this paper we prove that the cone $\PPD$ of positive, positive definite,discrete and strong almost periodic measures has an interesting property: givenany positive and positive definite measure $\mu$ smaller than some measure in$\PPD$, then the strong almost periodic part $\mu_S$ of $\mu$ is also in$\PPD$. We then use this result to prove that given a positive weighted comb$\omega$ with finite local complexity and pure point diffraction, any positivecomb less than $\omega$ has either trivial Bragg spectrum or a relatively denseset of Bragg peaks.

Positive Positive Definite Discrete Strong Almost Periodic Measures and Bragg Diffraction