## 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

## Abstract:

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 Ammannâ€“Beenker tilings.## References

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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: henna.koivusalo@bristol.ac.uk
**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: James.Walton@nottingham.ac.uk
- 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: https://doi.org/10.1090/tran/8633
- MathSciNet review: 4439500