Periodica Mathematica Hungarica

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

Continued fractions and irrational rotations

  • Naoto Shimaru
  • Keizo TakashimaEmail author


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.


Rational rotations Irrational rotations Ostrowski expansion Continued fraction expansion 

Mathematics Subject Classification

Primary 11K38 Secondary 11K31 11A55 



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.


  1. 1.
    M. Drmota, R.F. Tichy, in Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, vol. 1651 (Springer, Berlin, 1997)Google Scholar
  2. 2.
    G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Clarendon Press, Oxford, 1979)zbMATHGoogle Scholar
  3. 3.
    M. Iosifescu, C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications, vol. 547 (Kluwer Academic Publishers, Dordrecht, 2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    A. Khinchine, Metrische Kettenbruchprobleme. Compos. Math. 1, 361–382 (1935)MathSciNetzbMATHGoogle Scholar
  5. 5.
    A.Y. Khinchin, Continued Fractions (Dover Publications, New York, 1997)zbMATHGoogle Scholar
  6. 6.
    L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974)zbMATHGoogle Scholar
  7. 7.
    H. Niederreiter, Application of diophantine approximations to numerical integration, in Diophantine Approximation and Its Applications, ed. by C.F. Osgood (Academic Press, New York, 1973), pp. 129–199Google Scholar
  8. 8.
    A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen. Abh. Math. Sem. Univ. Hamburg 1, 77–98 (1922)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    J. Schoissengeier, On the discrepancy of \((n \alpha )\). Acta Arith. 44, 241–279 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    J. Schoissengeier, On the discrepancy of \((n \alpha )\), II. J. Number Theory 24, 54–64 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    J. Schoissengeier, An asymptotic expansion for \(\sum _{n\le N} \{ n\alpha +\beta \}\), in Number-theoretic analysis (Vienna, 1988–89), Lecture Notes in Mathematics, vol. 1452 (Springer, Berlin, 1990), pp. 199–205Google Scholar

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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