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 βZ^d for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.