Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-10T22:39:29.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Slow increasing functions and the largest partial quotients in continued fraction expansions

Published online by Cambridge University Press:  27 September 2016

JINHUA CHANG  and
HAIBO CHEN
JINHUA CHANG
Affiliation:
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, 430073, P. R. China. e-mails: jhchang7199@163.com; hiboo_chen@sohu.com
HAIBO CHEN*
Affiliation:
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei, 430073, P. R. China. e-mails: jhchang7199@163.com; hiboo_chen@sohu.com
*
†Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 1 - 14
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Barbolosi, D. and Faivre, C. Sur les grands quotients artiels du dévelopement en fraction continue. J. Number Theory 106 (2004), 299–325.Google Scholar
[2] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications (John Wiley & Sons, West Sussex, 1990).Google Scholar
[3] Fan, A. H., Liao, L. M., Wang, B. W. and Wu, J. On the fast khintchine spectrum in continued fractions. Monatsh. Math. 171 (2013), 329–340.Google Scholar
[4] Galambos, J. The distribution of the largest coefficient in continued fraction expansions. Quart. J. Math. Oxford 23 (1972), 147–151.Google Scholar
[5] Galambos, J. The largest coefficient in continued fractions and related problems. Proceedings of the conference on Diophantine Approximation (New York, 1973), 101–109.Google Scholar
[6] Galambos, J. An iterated logarithm type theorem for the largest coefficient in continued fractions. Acta Arith. 25 (1974), 359–364.Google Scholar
[7] Iosifescu, M. and Kraaikamp, C. Metrical theory of continued fractions. Math. Appl. vol. 547 (Kluwer Academic Publishers, Dordrecht, 2002).Google Scholar
[8] Jarník, V. Zur metrischen theorie der diophantischen approximationen. Prace Mat. Fiz. 36 (1929), 91–106.Google Scholar
[9] Jakimczuk, R. Functions of slow increase and integer sequences. J. Integer Seq. 13 (2010), Article 10.1.1.Google Scholar
[10] Jakimczuk, R. Integer sequences, functions of slow increase, and the Bell numbers. J. Integer Seq. 14 (2011), article 11.5.8.Google Scholar
[11] Jakimczuk, R. Functions of slow increase, integer sequences and conjectures. Int. Math. Forum 6 (2011), 2703–2709.Google Scholar
[12] Khintchine, A. Y. Continued Fractions (Translated by Groningen, Peter Wynn P.) (Noordhoff Ltd, 1963).Google Scholar
[13] Okano, T. Explicit continued fractions with expected partial quotient growth. Proc. Amer. Math. Soc. 130 (2002), 1603–1605.Google Scholar
[14] Philipp, W. A conjecture of Erdös on continued fraction. Acta Arith. 28 (1976), 379–386.Google Scholar
[15] Shang, Y. Functions of α-slow increase. Bull. Math. Anal. Appl. 4 (2012), 226–230.Google Scholar
[16] Wu, J. A remark on the growth of the denominators of convergents. Monatsh. Math. 147 (2006), 259–264.Google Scholar
[17] Wu, J. and Xu, J. The distribution of the largest digit in continued fraction expansions. Math. Proc. Camb. Phil. Soc. 146 (2009), 207–212.Google Scholar