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• 作者：Yoshiyuki Mori ; Keizo Takashima
• 关键词：Leading digits ; Irrational rotations ; Continued fractions ; \(\chi ^2\) ; statistic
• 刊名：Periodica Mathematica Hungarica
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：73
• 期：2
• 页码：224-239
• 全文大小：708 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2829
• 卷排序：73

文摘

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.