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

Abstract

We study statistics and frequencies of tiles in random incommensurable Kakutani sequences of partitions in R d and present new explicit formulas. Our results illustrate the dependence of the various frequencies on the underlying random substitution system solely in terms of its combinatorial structure, the volumes of the participating tiles and the entropy of the partitions, improving previous results concerning non-random Kakutani partitions and multiscale substitution tilings and extending them to random constructions. Based on the result above, we provide a formula for the gap distribution of Delone sets associated with multiscale substitution tilings of the real line.