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Uniform distribution of Kakutani partitions generated by substitution schemes

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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.

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Author notes

  1. Yotam Smilansky

    Present address: Current address: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

Authors and Affiliations

  1. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Yotam Smilansky

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  1. Yotam Smilansky
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Correspondence to Yotam Smilansky.

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Smilansky, Y. Uniform distribution of Kakutani partitions generated by substitution schemes. Isr. J. Math. 240, 667–710 (2020). https://doi.org/10.1007/s11856-020-2075-z

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  • DOI: https://doi.org/10.1007/s11856-020-2075-z

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