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


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.


Rational rotations Irrational rotations Continued fractions 

Mathematics Subject Classification

Primary 11K38 Secondary 11K31 11A55 



  1. 1.
    A. Bazarova, I. Berkes, L. Horváth, On the extremal theory of continued fractions. J. Theor. Prob. 29, 248–266 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Beck, Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting (Springer, NewYork, 2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    H. Behnke, Über die Verteilung von Irrationalitaten mod. 1. Abh. Math. Sem. Univ. Hamburg 1, 252–267 (1922)CrossRefzbMATHGoogle Scholar
  4. 4.
    K. Doi, N. Shimaru, K. Takashima, On upper estimates for discrepancies of irrational rotations: via rational rotation approximations. Acta Math. Hungr. 152(1), 109–113 (2017)CrossRefzbMATHGoogle Scholar
  5. 5.
    M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math. 1651, Springer, Berlin (1997)Google Scholar
  6. 6.
    G.H. Hardy, J.E. Littlewood, Some problems of Diophantine approximation: the lattice-points of a right-angled triangle. Abh. Math. Sem. Univ. Hamburg 1, 212–249 (1922)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Clarendon Press, Oxford, 1979)zbMATHGoogle Scholar
  8. 8.
    E. Hecke, Über analytische Functionen und die Verteilung von Zahlen mod. Eins, Abh. Math. Sem. Univ. Hamburg 1, 54–76 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Iosifescu, C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and its Applications, 547 (Kluwer Academic Pub, Dordrecht, 2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    A. Khintchine, Ein Satz über Kettenbruche, mit arithmetischen Anwendungen. Ann. Zeit. 18, 289–306 (1923)MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. Khintchine, Einige Sätze über Ketterbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92, 115–125 (1924)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Khinchine, Metrische Kettenbruchprobleme. Compositio Math. 1, 361–382 (1935)MathSciNetzbMATHGoogle Scholar
  13. 13.
    A.Ya. Khinchin, Continued Fractions (Dover Publications, New York, 1997)zbMATHGoogle Scholar
  14. 14.
    L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974)zbMATHGoogle Scholar
  15. 15.
    Y. Mori, K. Takashima, On the distribution of leading digits of \(a^n\): a study via \(\chi ^2\) statistics. Periodica Math. Hungr. 73, 224–239 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen. Abh. Math. Sem. Univ. Hamburg 1, 77–98 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    J. Schoissengeier, On the discrepancy of \((n \alpha )\). Acta Arith. 44, 241–279 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Schoissengeier, On the discrepancy of \((n \alpha )\), II. J. Number Theor. 24, 54–64 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. Schoissengeier, Eine explizite Formel für \(\sum _{n \ge N} B_2 (\{n \alpha \} )\), L.N.M. Springer, 1262, 134–138 (1987)Google Scholar
  21. 21.
    T. Setokuchi, On the discrepancy of irrational rotations with isolated large partial quotients: long term effects. Acta Math. Hungr. 147(2), 368–385 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    T. Setokuchi, K. Takashima, Discrepancies of irrational rotations with isolated large partial quotient. Unif. Distrib. Theory 2(9), 31–57 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    N. Shimaru, K. Takashima, On Discrepancies of Irrational Rotations: An Approach via Rational Rotations, Periodica Math. Hungr. (2016)Google Scholar
  24. 24.
    N. Shimaru, K. Takashima, Continued Fractions and Irrational Rotations. Acta Math. Hungr. 156(2), 449–458 (2016)CrossRefzbMATHGoogle Scholar
  25. 25.
    J. Vinson, Partial Sums of \(\zeta (\frac{1}{2})\) Modulo 1. Exp. Math. 10, 337–344 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations