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arXiv · 1602.08928

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

Michael Björklund·Tobias Hartnick·Felix Pogorzelski
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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.

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Michael Björklund, Tobias Hartnick, Felix Pogorzelski. 2016-02-29. Aperiodic order and spherical diffraction, I: Auto-correlation of model sets. https://doi.org/10.1112/plms.12091

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