  In this paper we study linearly repetitive Delone sets and prove, following
the work of Bellissard, Benedetti and Gambaudo, that the hull of a linearly
repetitive Delone set admits a properly nested sequence of box decompositions
(tower system) with strictly positive and uniformly bounded (in size and norm)
transition matrices. This generalizes a result of Durand for linearly recurrent
symbolic systems. Furthermore, we apply this result to give a new proof of a
classic estimation of Lagarias and Pleasants on the rate of convergence of
patch-frequencies.
