Linearly repetitive Delone sets are rectifiable
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 275-290.

We show that every linearly repetitive Delone set in the Euclidean d-space d , with d2, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice d . In the particular case when the Delone set X in d comes from a primitive substitution tiling of 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 β d for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

@article{AIHPC_2013__30_2_275_0,
     author = {Aliste-Prieto, Jos\'e and Coronel, Daniel and Gambaudo, Jean-Marc},
     title = {Linearly repetitive {Delone} sets are rectifiable},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {275--290},
     publisher = {Elsevier},
     volume = {30},
     number = {2},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.07.006},
     zbl = {1288.52011},
     mrnumber = {3035977},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.07.006/}
}
TY  - JOUR
AU  - Aliste-Prieto, José
AU  - Coronel, Daniel
AU  - Gambaudo, Jean-Marc
TI  - Linearly repetitive Delone sets are rectifiable
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
DA  - 2013///
SP  - 275
EP  - 290
VL  - 30
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2012.07.006/
UR  - https://zbmath.org/?q=an%3A1288.52011
UR  - https://www.ams.org/mathscinet-getitem?mr=3035977
UR  - https://doi.org/10.1016/j.anihpc.2012.07.006
DO  - 10.1016/j.anihpc.2012.07.006
LA  - en
ID  - AIHPC_2013__30_2_275_0
ER  - 
%0 Journal Article
%A Aliste-Prieto, José
%A Coronel, Daniel
%A Gambaudo, Jean-Marc
%T Linearly repetitive Delone sets are rectifiable
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 275-290
%V 30
%N 2
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2012.07.006
%R 10.1016/j.anihpc.2012.07.006
%G en
%F AIHPC_2013__30_2_275_0
Aliste-Prieto, José; Coronel, Daniel; Gambaudo, Jean-Marc. Linearly repetitive Delone sets are rectifiable. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 275-290. doi : 10.1016/j.anihpc.2012.07.006. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.07.006/

[1] J. Aliste-Prieto, D. Coronel, J.-M. Gambaudo, Rapid convergence to frequency for substitution tilings of the plane, Comm. Math. Phys. 306 no. 2 (2011), 365-380 | MR | Zbl

[2] D. Burago, B. Kleiner, Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps, Geom. Funct. Anal. 8 no. 2 (1998), 273-282 | MR | Zbl

[3] D. Burago, B. Kleiner, Rectifying separated nets, Geom. Funct. Anal. 12 no. 1 (2002), 80-92 | MR | Zbl

[4] N.G. De Bruijn, Algebraic theory of Penroseʼs nonperiodic tilings of the plane. I, Nederl. Akad. Wetensch. Indag. Math. 43 no. 1 (1981), 39-52 N.G. De Bruijn, Algebraic theory of Penroseʼs nonperiodic tilings of the plane. II, Nederl. Akad. Wetensch. Indag. Math. 43 no. 1 (1981), 53-66 | MR | Zbl

[5] D. Frettlöh, A. Garber, private communication.

[6] M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, vol. 2, Sussex, 1991, London Math. Soc. Lecture Note Ser. vol. 182, Cambridge Univ. Press, Cambridge (1993), 1-295 | MR

[7] B. Grünbaum, G.C. Shephard, Tilings and Patterns, W.H. Freeman and Company, New York (1989) | MR | Zbl

[8] R.A. Horn, Ch.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge (1990) | MR

[9] J.C. Lagarias, P.A.B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems 23 no. 3 (2003), 831-867 | MR | Zbl

[10] M. Laczkovich, Uniformly spread discrete sets in d , J. London Math. Soc. (2) 46 no. 1 (1992), 39-57 | MR | Zbl

[11] C.T. Mcmullen, Lipschitz maps and nets in Euclidean space, Geom. Funct. Anal. 8 (1998), 304-314 | MR | Zbl

[12] N. Priebe, B. Solomyak, Characterization of planar pseudo-self-similar tilings, Discrete Comput. Geom. 26 no. 3 (2001), 289-306 | MR | Zbl

[13] T. Rivière, D. Ye, Resolutions of the prescribed volume form equation, NoDEA 3 (1996), 323-369 | MR | Zbl

[14] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long range orientational order and no translational symmetry, Phys. Rev. Lett. 53 no. 20 (1984), 1951-1954

[15] Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math. 181 no. 1 (2011), 445-460 | MR | Zbl

[16] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom. 20 no. 2 (1998), 265-279 | MR | Zbl

[17] B. Solomyak, Pseudo-self-affine tilings in d , Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13 Zap. Nauchn. Sem. St.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 326 (2005), 198-213, J. Math. Sci. (N.Y.) 140 no. 3 (2007), 452-460 | EuDML | Zbl

[18] D. Toledo, Geometric group theory, 2: Asymptotic invariants of finite groups by M. Gromov, Bull. Amer. Math. Soc. 33 (1996), 395-398

Cited by Sources: