Cut and project sets with polytopal window II: Linear repetitivity

Koivusalo, Henna; Walton, James

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 Ammann–Beenker tilings.

Citation

Koivusalo, H., & Walton, J. (2022). Cut and project sets with polytopal window II: Linear repetitivity. Transactions of the American Mathematical Society, 375(7), 5097-5149. https://doi.org/10.1090/tran/8633

• Journal Article Type: Article
• Acceptance Date: Jan 3, 2022
• Online Publication Date: May 4, 2022
• Publication Date: Jul 1, 2022
• Deposit Date: Mar 4, 2022
• Publicly Available Date: May 4, 2022
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 375
• Issue: 7
• Pages: 5097-5149
• DOI: https://doi.org/10.1090/tran/8633
• Public URL: https://nottingham-repository.worktribe.com/output/7537030
• Publisher URL: https://www.ams.org/journals/tran/2022-375-07/S0002-9947-2022-08633-5/home.html