We show that every linearly repetitive Delone set in the Euclidean d-space , with , is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice . In the particular case when the Delone set X in comes from a primitive substitution tiling of , 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 for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.
Cite this article
Daniel Coronel, Jean-Marc Gambaudo, José Aliste-Prieto, Linearly repetitive Delone sets are rectifiable. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 275–290DOI 10.1016/J.ANIHPC.2012.07.006