Linearly repetitive Delone sets are rectifiable. (English) Zbl 1288.52011

Summary: We show that every linearly repetitive Delone set in the Euclidean \(d\)-space \(\mathbb R^d\), with \(d\geqslant 2\), is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice \(\mathbb Z^d\). In the particular case when the Delone set \(X\) in \(\mathbb R^d\) comes from a primitive substitution tiling of \(\mathbb R^d\), we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from \(X\) to the lattice \(\beta\mathbb Z^d\) for some positive \(\beta\). This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.


52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
Full Text: DOI arXiv


[1] Aliste-Prieto, J.; Coronel, D.; Gambaudo, J.-M., Rapid convergence to frequency for substitution tilings of the plane, Comm. Math. Phys., 306, 2, 365-380, (2011) · Zbl 1232.05049
[2] Burago, D.; Kleiner, B., Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal., 8, 2, 273-282, (1998) · Zbl 0902.26004
[3] Burago, D.; Kleiner, B., Rectifying separated nets, Geom. Funct. Anal., 12, 1, 80-92, (2002) · Zbl 1165.26007
[4] de Bruijn, N. G.; de Bruijn, N. G., Algebraic theory of penroseʼs nonperiodic tilings of the plane. II, Nederl. Akad. Wetensch. Indag. Math., Nederl. Akad. Wetensch. Indag. Math., 43, 1, 53-66, (1981) · Zbl 0457.05022
[5] D. Frettlöh, A. Garber, private communication.
[6] Gromov, M., Asymptotic invariants of infinite groups, (Geometric Group Theory, vol. 2, Sussex, 1991, London Math. Soc. Lecture Note Ser., vol. 182, (1993), Cambridge Univ. Press Cambridge), 1-295
[7] Grünbaum, B.; Shephard, G. C., Tilings and patterns, (1989), W.H. Freeman and Company New York · Zbl 0601.05001
[8] Horn, R. A.; Johnson, Ch. R., Matrix analysis, (1990), Cambridge University Press Cambridge, corrected reprint of the 1985 original · Zbl 0704.15002
[9] Lagarias, J. C.; Pleasants, P. A.B., Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23, 3, 831-867, (2003) · Zbl 1062.52021
[10] Laczkovich, M., Uniformly spread discrete sets in \(\mathbb{R}^d\), J. London Math. Soc. (2), 46, 1, 39-57, (1992) · Zbl 0774.11038
[11] McMullen, C. T., Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal., 8, 304-314, (1998) · Zbl 0941.37030
[12] Priebe, N.; Solomyak, B., Characterization of planar pseudo-self-similar tilings, Discrete Comput. Geom., 26, 3, 289-306, (2001) · Zbl 0997.52012
[13] Rivière, T.; Ye, D., Resolutions of the prescribed volume form equation, NoDEA, 3, 323-369, (1996) · Zbl 0857.35025
[14] Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. W., Metallic phase with long range orientational order and no translational symmetry, Phys. Rev. Lett., 53, 20, 1951-1954, (1984)
[15] Solomon, Y., Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math., 181, 1, 445-460, (2011) · Zbl 1217.52012
[16] Solomyak, B., Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20, 2, 265-279, (1998) · Zbl 0919.52017
[17] Solomyak, B., Pseudo-self-affine tilings in \(\mathbb{R}^d\), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13, Zap. Nauchn. Sem. St.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), J. Math. Sci. (N.Y.), 140, 3, 452-460, (2007), 282-283, translation in:
[18] Toledo, D., Geometric group theory, 2: asymptotic invariants of finite groups by M. Gromov, Bull. Amer. Math. Soc., 33, 395-398, (1996)
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.