# Scaling of diffraction intensities near the origin: Some rigorous results

## TL;DR

This paper rigorously analyzes how diffraction intensities scale near the origin in various ordered and disordered one-dimensional systems, providing insights into hyperuniformity and fluctuation behaviors.

## Contribution

It offers rigorous results on the scaling of diffraction intensities near the origin across different types of one-dimensional systems, including ordered, singular, and stochastic systems.

## Key findings

- Diffraction intensity scaling behavior characterized for various systems
- Hyperuniformity detection methods clarified through rigorous results
- Differences in diffraction spectra types elucidated

## Abstract

The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation behaviour known under the term hyperuniformity. Here, we consider one-dimensional systems with pure point, singular continuous and absolutely continuous diffraction spectra, which include perfectly ordered cut and project and inflation point sets as well as systems with stochastic disorder.