Acta Mathematica Hungarica
Volume/Issue:
Volume 125: Issue 3
Divergence points with fast growth orders of the partial quotients in continued fractions
Authors:
B. Wang Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074 China

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J. Wu Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074 China

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Pages:
261–274
Online Publication Date:
16 Sep 2009
Publication Date:
01 Nov 2009
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-009-9016-y
Keywords:
continued fractions; divergence point; Hausdorff dimension; 11A55; 28A80
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Abstract  

This paper is concerned with the divergence points with fast growth orders of the partial quotients in continued fractions. Let S be a nonempty interval. We are interested in the size of the set of divergence points

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$E_\varphi (S) = \left\{ {x \in [0,1):{\rm A}\left( {\frac{1} {{\varphi (n)}}\sum\limits_{k = 1}^n {\log a_k (x)} } \right)_{n = 1}^\infty = S} \right\},$$ \end{document}
where A denotes the collection of accumulation points of a sequence and φ: ℕ → ℕ with φ(n)/n → ∞ as n → ∞. Mainly, it is shown, in the case φ being polynomial or exponential function, that the Hausdorff dimension of Eφ(S) is a constant. Examples are also given to indicate that the above results cannot be expected for the general case.

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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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