  In this paper we prove that, for any integer $d>0$, every linearly repetitive
Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a
bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular
case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution
tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution
matrix which implies the existence of a homeomorphism with bounded displacement
from $X$ to the lattice lattice $\lambda\ZZ^d$ for some positive $\lambda$.
This condition includes primitive Pisot substitution tilings but also concerns
a much broader set of substitution tilings.
