Cut and project sets with polytopal window. II: Linear repetitivity. (English) Zbl 1491.52017
Summary: 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.
For Part I see [the authors, Ergodic Theory Dyn. Syst. 41, No. 5, 1431–1463 (2021; Zbl 1509.52019)].
For Part I see [the authors, Ergodic Theory Dyn. Syst. 41, No. 5, 1431–1463 (2021; Zbl 1509.52019)].
MSC:
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 52C45 | Combinatorial complexity of geometric structures |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
Keywords:
aperiodic order; cut and project sets; model sets; linear repetitivity; Diophantine approximationCitations:
Zbl 1509.52019References:
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