Aperiodic order and spherical diffraction, I: Auto-correlation of model sets

Abstract

We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hof's classical definition beyond the class of amenable groups, and provide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

Extended and revised version of the first part of the preprint previously circulated under the title "Aperiodic order and spherical diffraction", 36 pages

References in corpus (4)

• Pure Point Dynamical and Diffraction Spectra
• Dynamics on the graph of the torus parametrisation
• Ergodic properties of visible lattice points
• Pure point diffraction and Poisson summation

Cited by in corpus (7)

• Approximate lattices
• Leptin densities in amenable groups
• Borel density for approximate lattices
• Closed approximate subgroups: compactness, amenability and approximate lattices
• Linear repetitivity beyond abelian groups
• Classifying the Dynamics of Architected Materials by Groupoid Methods
• Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao