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On the distribution of the leading digit of \(a^n\): a study via \(\chi ^2\) statistics

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Abstract

We investigate irrational rotations with isolated large partial quotients from the point of view of the distribution of the leading digit of \(a^n\). We prove some mathematical formulae explaining the unusual behavior of the \(\chi ^2\) statistic of the leading digits of \(a^n\), where \(\log _{10}a\) has a single isolated large partial quotient in its continued fraction expansion. We also report that hills appear infinitely often in the graphs of \(\chi ^2\) statistics and that there are many different types of shapes of hills.

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Acknowledgments

The authors would like to express their hearty thanks to Prof. Dalibor Volny for his helpful advice and fruitful discussions on our problems. Without his support, this paper would not have been accomplished. Their thanks are also dedicated to the referee and the editor, for their valuable and important comments.

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Authors and Affiliations

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

    Yoshiyuki Mori & Keizo Takashima

Authors
  1. Yoshiyuki Mori
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  2. Keizo Takashima
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Correspondence to Keizo Takashima.

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Mori, Y., Takashima, K. On the distribution of the leading digit of \(a^n\): a study via \(\chi ^2\) statistics. Period Math Hung 73, 224–239 (2016). https://doi.org/10.1007/s10998-016-0138-z

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