Rapid convergence to frequency for Substitution Tilings of the Plane

TL;DR

This paper investigates how quickly the frequency of specific tiles in self-similar plane tilings converges, providing estimates of oscillation based on the boundary shape, applicable to many known tilings.

Contribution

It introduces bounds on tile occurrence oscillations in self-similar tilings, extending understanding of frequency convergence in these structures.

Findings

Provides estimates of oscillation depending only on the Jordan curve

Applicable to most known self-similar tilings

Shows rapid convergence to tile frequency

Abstract

This paper concerns self-similar tilings in dimension 2. 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 most 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.