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Statistics of patterns in typical cut and project sets

Published online by Cambridge University Press:  13 March 2018

ALAN HAYNES ,
ANTOINE JULIEN ,
HENNA KOIVUSALO  and
JAMES WALTON
ALAN HAYNES
Affiliation:
Department of Mathematics, University of Houston, Philip Guthrie Hoffman Hall, 3551 Cullen Blvd., Room 641, Houston, TX 77204-3008, USA email [email protected]
ANTOINE JULIEN
Affiliation:
Nord universitet, Levanger Røstad, Høgskoleveien 27, 7600 Levanger, Norway email [email protected]
HENNA KOIVUSALO
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-platz 1, 1090 Vienna, Austria email [email protected]
JAMES WALTON
Affiliation:
Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham DH1 3LE, UK email [email protected]
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Abstract

In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies. We also give bounds for repetitivity and repulsivity functions. The proofs use ideas and tools developed in discrepancy theory.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3365 - 3387
Copyright
© Cambridge University Press, 2018 

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References

Adamczewski, B.. Symbolic discrepancy and self-similar dynamics. Ann. Inst. Fourier (Grenoble) 54(7) (2004), 2201–2234.Google Scholar
Aliste-Prieto, J., Coronel, D. and Gambaudo, J.-M.. Rapid convergence to frequency for substitution tilings of the plane. Comm. Math. Phys. 306(2) (2011), 365–380.Google Scholar
Arnoux, P., Mauduit, C., Shiokawa, I. and Tamura, J.-I.. Complexity of sequences defined by billiard in the cube. Bull. Soc. Math. France 122(1) (1994), 1–12.Google Scholar
Baryshnikov, Y.. Complexity of trajectories in rectangular billiards. Comm. Math. Phys. 174(1) (1995), 43–56.Google Scholar
Beresnevich, V., Dickinson, D. and Velani, S.. Measure theoretic laws for lim sup sets. Mem. Amer. Math. Soc. 179(846) (2006), x+91.Google Scholar
Berthé, V. and Vuillon, L.. Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discrete Math. 223(1–3) (2000), 27–53.Google Scholar
Bufetov, A. I. and Solomyak, B.. Limit theorems for self-similar tilings. Comm. Math. Phys. 319(3) (2013), 761–789.Google Scholar
Cassels, J. W. S.. An Introduction to Diophantine Approximation (Cambridge Tracts in Mathematics and Mathematical Physics, 45) . Cambridge University Press, New York, 1957.Google Scholar
Dreher, F., Kesseböhmer, M., Mosbach, A., Samuel, T. and Steffens, M.. Regularity of aperiodic minimal subshifts. Bull. Math. Sci. (Mar 2017), doi:10.1007/s13373-017-0102-0.Google Scholar
Fogg, N. P.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794) . Eds. Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A.. Springer, Berlin, 2002.Google Scholar
Forrest, A., Hunton, J. and Kellendonk, J.. Topological invariants for projection method patterns. Mem. Amer. Math. Soc. 159(758) (2002), x+120.Google Scholar
Grepstad, S. and Lev, N.. Sets of bounded discrepancy for multi-dimensional irrational rotation. Geom. Funct. Anal. 25(1) (2015), 87–133.Google Scholar
Gröger, M., Kesseböhmer, M., Mosbach, A., Samuel, T. and Steffens, M.. A classification of aperiodic order via spectral metrics & Jarník sets, Preprint, 2016, arXiv:1601.06435.Google Scholar
Harman, G.. Metric Number Theory (London Mathematical Society Monographs. New Series, 18) . The Clarendon Press, Oxford University Press, New York, 1998.Google Scholar
Haynes, A., Koivusalo, H. and Walton, J.. A characterization of linearly repetitive cut and project sets. Nonlinearity 31(2) (2018), 515–539.Google Scholar
Haynes, A., Koivusalo, H., Walton, J. and Sadun, L.. Gaps problems and frequencies of patches in cut and project sets. Math. Proc. Cambridge Philos. Soc. 161(1) (2016), 65–85.Google Scholar
Julien, A.. Complexity as a homeomorphism invariant for tiling spaces. Ann. Inst. Fourier (Grenoble) 67(2) (2017), 539–577.Google Scholar
Kesten, H.. On a conjecture of Erdős and Szüsz related to uniform distribution mod 1. Acta Arith. 12 (1966/1967), 193–212.Google Scholar
Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (Pure and Applied Mathematics) . Wiley-Interscience [John Wiley & Sons], New York, London, Sydney, 1974.Google Scholar
Laczkovich, M.. Uniformly spread discrete sets in R d . J. Lond. Math. Soc. (2) 46(1) (1992), 39–57.Google Scholar
Lagarias, J. C. and Pleasants, P. A. B.. Local complexity of Delone sets and crystallinity. Canad. Math. Bull. 45(4) (2002), 634–652; dedicated to Robert V. Moody.Google Scholar
Matei, B. and Meyer, Y.. Simple quasicrystals are sets of stable sampling. Complex Var. Elliptic Equ. 55(8–10) (2010), 947–964.Google Scholar
Moody, R. V.. Meyer sets and their duals. The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995) (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 489) . Kluwer, Dordrecht, 1997, pp. 403–441.Google Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics. Amer. J. Math. 60(4) (1938), 815–866.Google Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), 1–42.Google Scholar
Sadun, L.. Exact regularity and the cohomology of tiling spaces. Ergod. Th. & Dynam. Sys. 31(6) (2011), 1819–1834.Google Scholar
Savinien, J.. A metric characterisation of repulsive tilings. Discrete Comput. Geom. 54(3) (2015), 705–716.Google Scholar
Shechtman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984), 1951–1953.Google Scholar
Solomyak, B.. Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2) (1998), 265–279.Google Scholar