Linearly repetitive Delone sets are rectifiable

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 beta. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings. (C) 2012 Elsevier Masson SAS. All rights reserved.