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

13 pages, 3 figures. The abstract has been changed. To appear in Journal of Mathematical Analysis and Applications

References in corpus (3)

• Dynamical systems arising from random substitutions
• Random Substitution Tilings and Deviation Phenomena
• Multiscale Substitution Tilings

Cited by in corpus (1)

• Discrepancy and rectifiability of almost linearly repetitive Delone sets