Kinematic Diffraction from a Mathematical Viewpoint

Abstract

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.

30 pages, invited review

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Cited by in corpus (21)

• Generalized stealthy hyperuniform processes : maximal rigidity and the bounded holes conjecture
• Entropy and diffraction of the $k$-free points in $n$-dimensional lattices
• Mathematical diffraction of aperiodic structures
• Spectral and topological properties of a family of generalised Thue-Morse sequences
• Dynamical versus diffraction spectrum for structures with finite local complexity
• On the Fourier Analysis of Measures with Meyer Set Support
• Aperiodic crystals and beyond
• On the Entropy of a Family of Random Substitutions
• A short guide to pure point diffraction in cut-and-project sets
• Squirals and beyond: Substitution tilings with singular continuous spectrum
• On Weighted Dirac Combs Supported Inside Model Sets
• Examples of substitution systems and their factors
• On the notions of symmetry and aperiodicity for Delone sets
• The inverse problem of pure point diffraction -- examples and open questions
• A comment on the relation between diffraction and entropy
• Positive Positive Definite Discrete Strong Almost Periodic Measures and Bragg Diffraction
• A note on measures vanishing at infinity
• Note on the set of Bragg peaks with high intensity
• Diffraction of return time measures
• Squiral diffraction
• Recent progress in mathematical diffraction