Linearly repetitive Delone sets are rectifiable
Annales de l'I.H.P. Analyse non linéaire, Tome 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.

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     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},
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Aliste-Prieto, José; Coronel, Daniel; Gambaudo, Jean-Marc. Linearly repetitive Delone sets are rectifiable. Annales de l'I.H.P. Analyse non linéaire, Tome 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/

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