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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Cut and project sets with polytopal window II: linear repetitivity
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by Henna Koivusalo and James J. Walton PDF
Trans. Amer. Math. Soc. 375 (2022), 5097-5149 Request permission


In this paper we give a complete characterisation of linear repetitivity for cut and project schemes with convex polytopal windows satisfying a weak homogeneity condition. This answers a question of Lagarias and Pleasants from the 90s for a natural class of cut and project schemes which is large enough to cover almost all such polytopal schemes which are of interest in the literature. We show that a cut and project scheme in this class has linear repetitivity exactly when it has the lowest possible patch complexity and satisfies a Diophantine condition. Finding the correct Diophantine condition is a major part of the work. To this end we develop a theory, initiated by Forrest, Hunton and Kellendonk, of decomposing polytopal cut and project schemes to factors. We also demonstrate our main theorem on a wide variety of examples, covering all classical examples of canonical cut and project schemes, such as Penrose and Ammann–Beenker tilings.
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Additional Information
  • Henna Koivusalo
  • Affiliation: School of Mathematics, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom
  • MR Author ID: 1062599
  • Email:
  • James J. Walton
  • Affiliation: School of Mathematical Sciences, Mathematical Sciences Building, University Park, Nottingham NG7 2RD, United Kingdom
  • MR Author ID: 1162597
  • ORCID: 0000-0003-3804-444X
  • Email:
  • Received by editor(s): May 18, 2021
  • Received by editor(s) in revised form: December 23, 2021, and January 3, 2022
  • Published electronically: May 4, 2022
  • Additional Notes: We gratefully acknowledge the support of the London Mathematical Society, Scheme 2. The research of the second author was partially supported by EPSRC grant EP/R013691/1.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 5097-5149
  • MSC (2020): Primary 52C23; Secondary 52C45
  • DOI:
  • MathSciNet review: 4439500