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.
Similar content being viewed by others
References
M. Drmota, R.F. Tichy, in Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, vol. 1651 (Springer, Berlin, 1997)
G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Clarendon Press, Oxford, 1979)
M. Iosifescu, C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications, vol. 547 (Kluwer Academic Publishers, Dordrecht, 2002)
A. Khinchine, Metrische Kettenbruchprobleme. Compos. Math. 1, 361–382 (1935)
A.Y. Khinchin, Continued Fractions (Dover Publications, New York, 1997)
L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences (Wiley, New York, 1974)
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. 129–199
A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen. Abh. Math. Sem. Univ. Hamburg 1, 77–98 (1922)
J. Schoissengeier, On the discrepancy of \((n \alpha )\). Acta Arith. 44, 241–279 (1984)
J. Schoissengeier, On the discrepancy of \((n \alpha )\), II. J. Number Theory 24, 54–64 (1986)
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. 199–205
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shimaru, N., Takashima, K. Continued fractions and irrational rotations. Period Math Hung 75, 155–158 (2017). https://doi.org/10.1007/s10998-016-0175-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-016-0175-7