Scaling of diffraction intensities near the origin: Some rigorous results

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.

arXiv comment24 pages

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• Characterizing the Hyperuniformity of Ordered and Disordered Two-Phase Media
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
• Superconducting properties of Fibonacci chains with enhanced superconducting pairing at the boundaries
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• Scaling of the diffraction measure of $k$-free integers near the origin
• On a family of singular continuous measures related to the doubling map
• Asymptotics of the $k$-free diffraction measure via discretisation