Almost periodic measures and Bragg diffraction

In this paper we prove that the cone PDS(G) of positive, positive definite, discrete and strong almost periodic measures over a σ-compact, locally compact Abelian group G has an interesting property: given any positive and positive definite measure μ smaller than some measure in PDS(G), the strong almost periodic part μS of μ is also in PDS(G). We then use this result to prove that given a positive-weighted Dirac comb ω with finite local complexity and pure point diffraction, any positive Dirac comb less than ω has either a trivial Bragg spectrum or a relatively dense set of Bragg peaks. (paper)