Uniform Distribution of Kakutani Partitions Generated By Substitution Schemes
Abstract
Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. Using our recent path counting results for directed weighted graphs, we show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.
Keywords
Cite
@article{arxiv.1805.02213,
title = {Uniform Distribution of Kakutani Partitions Generated By Substitution Schemes},
author = {Yotam Smilansky},
journal= {arXiv preprint arXiv:1805.02213},
year = {2020}
}
Comments
25 pages, 22 figures; revision after the referee report, to appear in Israel Journal of Mathematics. Results unchanged, but substantial changes in organization of the paper; details and references added