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Cut and project sets with polytopal window I: Complexity

Part of: Discrete geometry

Published online by Cambridge University Press:  20 February 2020

HENNA KOIVUSALO  and
JAMES J. WALTON Open the ORCID record for JAMES J. WALTON [Opens in a new window]
HENNA KOIVUSALO
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-platz 1, 1090Wien, Austria email henna.koivusalo@univie.ac.at
JAMES J. WALTON
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ, UK email Jamie.Walton@glasgow.ac.uk
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Abstract

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We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

Keywords

aperiodic ordercut and projectmodel setscomplexity

MSC classification

Primary: 52C23: Quasicrystals, aperiodic tilings
Secondary: 52C45: Combinatorial complexity of geometric structures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1431 - 1463
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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