Positive Positive Definite Discrete Strong Almost Periodic Measures and Bragg Diffraction

In this paper we prove that the cone $\PPD$ of positive, positive definite, discrete and strong almost periodic measures has an interesting property: given any 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 positive comb less than $\omega$ has either trivial Bragg spectrum or a relatively dense set of Bragg peaks.