Linearly repetitive Delone sets are rectifiable

Abstract

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