Cut and project sets with polytopal window II: linear repetitivity

Koivusalo, Henna; Walton, James

doi:10.1090/tran/8633

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 Ammann–Beenker tilings.

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