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On the discrepancy of some generalized Kakutani's sequences of partitions

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2012

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Drmota, Michael

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Uniform Distribution Theory. Mathematical Institute of the Slovak Academy of Sciences. 2012, 7(1), pp. 75-104. eISSN 1336-913X

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In this paper we study a class of generalized Kakutani’s sequences of partitionsof [0,1],constructedbyusingthetechniqueofsuccessive ρ−refinements. Our main focus is to derive bounds for the discrepancy of these sequences. The approach that we use is based on a tree representation of the sequence of partitions which is precisely the parsing tree generated by Khodak’s coding algorithm. With the help of this technique we derive (partly up to a logarithmic factor) optimal upper bound in the so-called rational case. The upper bounds in the irrational case that we obtain are weaker, since they heavily depend on Diophantine approximation properties of a certain irrational number. Finally, we present an application of these results to a class of fractals.

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510 Mathematik

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ISO 690DRMOTA, Michael, Maria INFUSINO, 2012. On the discrepancy of some generalized Kakutani's sequences of partitions. In: Uniform Distribution Theory. Mathematical Institute of the Slovak Academy of Sciences. 2012, 7(1), pp. 75-104. eISSN 1336-913X
BibTex
@article{Drmota2012discr-49646,
  year={2012},
  title={On the discrepancy of some generalized Kakutani's sequences of partitions},
  url={https://math.boku.ac.at/udt/vol07/no1/04DrmInf.pdf},
  number={1},
  volume={7},
  journal={Uniform Distribution Theory},
  pages={75--104},
  author={Drmota, Michael and Infusino, Maria}
}
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URL der Originalveröffentl.
https://math.boku.ac.at/udt/vol07/no1/04DrmInf.pdf

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2020-05-26

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