Positive Positive Definite Discrete Strong Almost Periodic Measures and Bragg Diffraction

TL;DR

This paper investigates properties of positive, positive definite, discrete, and strong almost periodic measures, showing how their structure influences Bragg diffraction patterns in weighted point sets with finite local complexity.

Contribution

It proves that the strong almost periodic part of certain measures remains within the same class and links measure properties to the nature of Bragg spectra in diffraction.

Findings

The strong almost periodic part of measures smaller than a measure in PPD is also in PPD.

Measures less than a pure point diffractive measure have either trivial or dense Bragg spectra.

The results connect measure structure with diffraction pattern characteristics.

Abstract

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.