Abstract

The notion of linearly recurrent subshift has been introduced in [Du, DHS] to study the relations between the substitutive dynamical systems and the stationary dimension groups. In an independent way, the similar notion of linearly repetitive Delone sets of the Euclidean space R^d appears in [LP1]. For a Delone set X of R^d, the repetitivity function MX(R) is the least M (possibly infinite) such that every closed ball B of radius M intersected with X contains a translated copy of any patch with diameter smaller than 2R. A Delone set X is said linearly repetitive if there exists a constant L such that MX(R) < LR for all R > 0.