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On the \(\chi ^2\) statistics of leading digits of irrational rotations with a large first or second partial quotient

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Abstract

We derive exact formulas for the \(\chi ^2\) statistics of the distribution of the leading digit of \(a^n\), where \(\log _{10} a\) has a large first or second partial quotient in its continued fraction expansion. We also give a mathematical explanation for the difference between the behavior of the discrepancy and that of the distribution of leading digits.

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Authors and Affiliations

  1. Department of Applied Mathematics, Graduate School of Science, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700-0005, Japan

    Toshifumi Nagayoshi

  2. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700-0005, Japan

    Keizo Takashima

Authors
  1. Toshifumi Nagayoshi
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  2. Keizo Takashima
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Correspondence to Keizo Takashima.

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Nagayoshi, T., Takashima, K. On the \(\chi ^2\) statistics of leading digits of irrational rotations with a large first or second partial quotient. Period Math Hung 80, 158–171 (2020). https://doi.org/10.1007/s10998-019-00306-0

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  • DOI: https://doi.org/10.1007/s10998-019-00306-0

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