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Periodica Mathematica Hungarica

, Volume 75, Issue 2, pp 155–158 | Cite as

Continued fractions and irrational rotations

  • Naoto Shimaru
  • Keizo TakashimaEmail author
Article

Abstract

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.

Keywords

Rational rotations Irrational rotations Ostrowski expansion Continued fraction expansion 

Mathematics Subject Classification

Primary 11K38 Secondary 11K31 11A55 

Notes

Acknowledgements

The authors would like to express their hearty thanks to the referee and the editor for their valuable and important comments, which improved the first version of the paper.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

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

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