Uniform Distribution of Kakutani Partitions Generated By Substitution Schemes

TL;DR

This paper extends Kakutani's classical result by demonstrating that a broad family of tile partitions generated by substitution schemes, including multiple scales, are uniformly distributed, with implications for tile frequency and volume distribution.

Contribution

It introduces a new class of substitution-generated partitions with multiple scales and proves their uniform distribution using path counting in weighted graphs.

Findings

Sequences of partitions are uniformly distributed.

Limiting frequencies relate to tile types and volumes.

Results extend Kakutani's original theorem.

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.