Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings

TL;DR

This paper analyzes the statistical properties of tiles in random Kakutani partitions and multiscale substitution tilings, providing explicit formulas that relate structure, volume, and entropy, and extending previous results to random systems.

Contribution

It introduces explicit formulas for tile statistics in random Kakutani sequences, advancing understanding of their dependence on combinatorial and entropy factors.

Findings

Explicit formulas for tile statistics depending on structure and entropy

Improved results over non-random Kakutani partitions and tilings

Gap distribution formula for Delone sets in multiscale tilings

Abstract

We study statistics of tiles in random incommensurable Kakutani sequences of partitions in $\mathbb{R}^d$. We provide explicit formulas that illustrate the dependence on the combinatorial structure, the volumes of the participating tiles and the entropy of the partitions in the underlying random substitution system. These improve previous results for non-random Kakutani partitions and multiscale substitution tilings, and imply a gap distribution formula for Delone sets associated with multiscale substitution tilings of the real line.