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ASYMPTOTIC BEHAVIOUR FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Part of: Classical measure theory Real functions Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  18 April 2024

XIAO CHEN Open the ORCID record for XIAO CHEN [Opens in a new window] ,
LULU FANG Open the ORCID record for LULU FANG [Opens in a new window] ,
JUNJIE LI Open the ORCID record for JUNJIE LI [Opens in a new window] ,
LEI SHANG Open the ORCID record for LEI SHANG [Opens in a new window]  and
XIN ZENG Open the ORCID record for XIN ZENG [Opens in a new window]
XIAO CHEN
Affiliation:
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: 469706659@qq.com
LULU FANG
Affiliation:
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: fanglulu@njust.edu.cn
JUNJIE LI
Affiliation:
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: 2233504975@qq.com
LEI SHANG*
Affiliation:
College of Sciences, Nanjing Agricultural University, Nanjing 210095, China
XIN ZENG
Affiliation:
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China e-mail: very.zeng@outlook.com
*
e-mail: auleishang@gmail.com
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Abstract

Let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of an irrational number $x\in [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all $x\in [0,1)$,

$$ \begin{align*} \liminf_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n} = 0\quad \text{and}\quad \limsup_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n}=1. \end{align*} $$

We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.

Keywords

continued fractionsproduct of consecutive partial quotientsresidual setsHausdorff dimension

MSC classification

Primary: 11K50: Metric theory of continued fractions
Secondary: 26A21: Classification of real functions; Baire classification of sets and functions 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The research is supported by the National Natural Science Foundation of China (No. 11801591), the Natural Science Foundation of Jiangsu Province (No. BK20231452), the Fundamental Research Funds for the Central Universities (No. 30922010809) and the China Postdoctoral Science Foundation (No. 2023M731697).

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