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 $\beta$. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.