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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.
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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 

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References

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