Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d-space Rd, with d⩾2, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Zd. In the particular case when the Delone set X in Rd comes from a primitive substitution tiling of Rd, 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 βZd for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.