Abstract
An estimate for dispersion of lengths of continued fractions is proved for fixed denominator. This estimate improves the trivial one by the logarithm of the denominator.
Similar content being viewed by others
References
I. V. Arnold, Number Theory [in Russian], Uchpedgiz (1939).
V. I. Arnold, Continued Fractions [in Russian], Moscow (2000).
Arnold’s Problems [in Russian], Fazis, Moscow (2000).
M. O. Avdeeva, “On the statistics of partial quotients of finite continued fractions,” Funkts. Anal. Prilozhen., 38, No. 2, 1–11 (2004).
M. O. Avdeeva and V. A. Bykovskii, A Solution of Arnold’s Problem on the Gauss-Kuzmin Statistic, preprint FEB RAS Khabarovsk Division of the Institute for Applied Mathematics, No. 8, Dalnauka, Vladivostok (2002).
V. Baladi and B. Valle, “Euclidean algorithms are Gaussian,” J. Number Theory, 110, No. 2, 331–386 (2005).
J. D. Dixon, “The number of steps in the Euclidean algorithm,” J. Number Theory, 2, 414–422 (1970).
H. Heilbronn, “On the average length of a class of finite continued fractions,” in: Abhandlungen aus Zahlentheorie und Analysis, VEB Deutsher Verlag der Wissenschaften, Berlin, Plenum Press, New York (1968), pp. 89–96.
D. Hensley, “The number of steps in the Euclidean algorithm,” J. Number Theory, 49, No. 2, 142–182 (1994).
A. Ya. Khinchin, Continued Fractions [in Russian], Fizmatgiz, Moscow (1961).
J. W. Porter, “On a theorem of Heilbronn,” Mathematika, 22, no. 1, 20–28 (1975).
A. V. Ustinov, “On statistic properties of continued fractions,” in: Works on Number Theory [in Russian], Zap. Nauchn. Sem. St. Peterb. Otd. Mat. Inst. im. V. A. Steklova, Ross. Akad. Nauk (POMI), Vol. 322, St. Petersburg (2005), pp. 186–211.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 15–26, 2005.
Rights and permissions
About this article
Cite this article
Bykovskii, V.A. Estimate for dispersion of lengths of continued fractions. J Math Sci 146, 5634–5643 (2007). https://doi.org/10.1007/s10958-007-0378-9
Issue Date:
DOI: https://doi.org/10.1007/s10958-007-0378-9