Tower systems for Linearly repetitive Delone sets

TL;DR

This paper investigates linearly repetitive Delone sets, establishing the existence of tower systems with bounded transition matrices, and applies this to refine convergence rate estimates of patch frequencies.

Contribution

It generalizes Durand's result to Delone sets and provides a new proof for convergence rate estimates of patch frequencies.

Findings

Existence of tower systems with positive, bounded transition matrices

Generalization of Durand's result to Delone sets

New proof of convergence rate of patch frequencies

Abstract

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.