Periodica Mathematica Hungarica

, Volume 73, Issue 2, pp 224–239 | Cite as

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

  • Yoshiyuki Mori
  • Keizo Takashima


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.


Leading digits Irrational rotations Continued fractions \(\chi ^2\) statistic 

Mathematics Subject Classification

Primary 11K38 Secondary 11K31 11A55 



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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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsOkayama University of ScienceOkayamaJapan

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