Rapid Convergence to Frequency for Substitution Tilings of the Plane

Abstract

This paper concerns self-similar tilings of the Euclidean plane. We consider the number of occurrences of a given tile in any domain bounded by a Jordan curve. For a large class of self-similar tilings, including many well-known examples, we give estimates of the oscillation of this number of occurrences around its average frequency times the total number of tiles in the domain, which depend only on the Jordan curve.Date: November 12, 2018.

Limit Theorems for Self-Similar Tilings

“…Aliste-Prieto, Coronel and Gambaudo [2,3] obtain analogous deviation estimates. The paper [2], which deals with the d = 2 case, estimates the deviation of average from the frequency for general Jordan domains and for very general substitution tilings, including non-FLC tilings, the "pinwheel-like" tilings and tiles with fractal boundary. However, the extension to d > 2 in [3] handles only the case of "small" θ 2 under the stronger assumption |θ 2 | ≤ θ 1 d 1 .…”

“…These deviation bounds are sharp, at least, in the general case. There are related recent results by Solomon [32,33] and Aliste-Prieto, Coronel, Gambaudo [2,3], who obtained estimates for the rate of convergence to frequency of the number of prototiles per volume for a class of domains. They were motivated by questions on bi-Lipschitz equivalence and bounded displacement of separated nets, arising from self-similar tilings, to the lattice.…”

“…It is an efficient hierarchical "packing" of a Lipschitz domain by tiles of varying orders. Analogous constructions have been used in [32,33,2,3,31].…”

Linearly repetitive Delone sets are rectifiable

“…Notice that the substitution tilings we consider in this paper are locally finite. A simple computation yields (see [ACG11] for details): Lemma 6.2. Let d ≥ 2 and T be a locally finite tiling of R d , then each ball of radius smaller than or equal to 2R T meets at most K T tiles of T .…”

“…It follows from Lemma 5.4 that the sequence of bi-Lipschitz constants of the bi-Lipschitz homeomorphisms Ψn ,m is bounded and thus, using Azerlà-Ascoli theorem, there exists an accumulation point Φ which is a bi-Lipschitz homeomorphism on We first state Theorem 6.1 below, which is a generalization of the main theorem in [ACG11]. Theorem 6.1.…”