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Periodica Mathematica Hungarica

, Volume 78, Issue 1, pp 88–97 | Cite as

On the distribution of partial sums of irrational rotations

  • Yoshiyuki Mori
  • Naoto Shimaru
  • Keizo TakashimaEmail author
Article
  • 70 Downloads

Abstract

For an irrational \(\alpha \), we investigate the sums \(\sum _{i=1}^n \left( \{i \alpha \} - \frac{1}{2} \right) \) and \(\sum _{i=1}^n \left\{ \left( \{i \alpha \} - \frac{1}{2} \right) ^2 - \frac{1}{12} \right\} \). We discuss exact formulae and asymptotic estimates for these sums and point out interesting geometrical properties of their graphs in the case when the continued fraction expansion of \(\alpha \) has a large isolated partial quotient.

Keywords

Rational rotations Irrational rotations Continued fractions 

Mathematics Subject Classification

Primary 11K38 Secondary 11K31 11A55 

Notes

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsOkayama University of ScienceOkayamaJapan
  2. 2.Department of Applied Mathematics, Graduate School of ScienceOkayama University of ScienceOkayamaJapan

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