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

We show that every linearly repetitive Delone set in the Euclidean d-space ${ℝ}^{d}$, with $d⩾2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice ${ℤ}^{d}$. In the particular case when the Delone set X in ${ℝ}^{d}$ comes from a primitive substitution tiling of ${ℝ}^{d}$, 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 $\beta {ℤ}^{d}$ 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},
publisher = {Elsevier},
volume = {30},
number = {2},
year = {2013},
pages = {275-290},
doi = {10.1016/j.anihpc.2012.07.006},
zbl = {1288.52011},
mrnumber = {3035977},
language = {en},
url = {http://www.numdam.org/item/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://www.numdam.org/item/AIHPC_2013__30_2_275_0/

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