zbMATH — the first resource for mathematics

Substitution tilings and separated nets with similarities to the integer lattice. (English) Zbl 1217.52012
Summary: We show that any primitive substitution tiling of \(\mathbb R^2\) creates a separated net which is biLipschitz to \(\mathbb Z^{2}\). Then we show that if \(H\) is a primitive Pisot substitution in \(\mathbb R^d\), for every separated net \(Y\), that corresponds to some tiling \(\tau \in X_H\), there exists a bijection \(\Phi \) between \(Y\) and the integer lattice such that \(\sup_{y\in Y}\|\Phi (y)-y\|<\infty\). As a corollary, we get that we have such a \(\Phi \) for any separated net that corresponds to a Penrose tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
Full Text: DOI arXiv
[1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. · Zbl 0484.15016
[2] D. Burago and B. Kleiner, Separated nets in Euclidean space and Jacobians of biLipschitz map, Geometric and Functional Analysis 8 (1998), 273–282. · Zbl 0902.26004
[3] D. Burago and B. Kleiner, Rectifying separated nets, Geometric and Functional Analysis 12 (2002), 80–92. · Zbl 1165.26007
[4] W. A. Deuber, M. Simonovits and V. T. Sos, A note on paradoxical metric spaces, Studia Scientiarum Mathematicarum Hungarica 30 (1995), 17–23. · Zbl 0857.54030
[5] M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory Vol. II (G. Niblo and M. Roller, eds.), London Mathematical Society Lecture Note Series 182, Cambridge University Press, Cambridge, 1993. · Zbl 0841.20039
[6] B. Grunbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Co., New York, 1987.
[7] L. Hogben (Ed.), Handbook of Linear Algebra, Chapman and Hall/CRC Press, 2007. · Zbl 1122.15001
[8] M. Laczkovich, Uniformly spread discrete sets in \(\mathbb{R}\)d, Journal of the London Mathematical Society. Second Series 46 (1992), 39–57. · Zbl 0774.11038
[9] C. T. McMullen, Lipschitz maps and nets in Euclidean space, Geometric and Functional Analysis 8 (1998), 304–314. · Zbl 0941.37030
[10] M. Queffelec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Mathematics 1294, Springer-Verlag, Berlin, 1987.
[11] C. Radin, Miles of Tiles, American Mathematical Society, Providence, RI, 1999. · Zbl 0932.52005
[12] E. A. Robinson, Jr., Symbolic dynamics and tilings of \(\mathbb{R}\)n, Proceedings of Symposia in Applied Mathematics 60 (2004), 81–119. · Zbl 1076.37010
[13] Y. Solomon, The net created from the Penrose Tiling is biLipschitz to the integer lattice, arXiv:0711.3707v1 (2008).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.