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ON THE INCREASING PARTIAL QUOTIENTS OF CONTINUED FRACTIONS OF POINTS IN THE PLANE

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

Published online by Cambridge University Press:  17 December 2021

MEIYING LÜ Open the ORCID record for MEIYING LÜ [Opens in a new window]  and
ZHENLIANG ZHANG Open the ORCID record for ZHENLIANG ZHANG [Opens in a new window]
MEIYING LÜ
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, PR China e-mail: lmy19831102@163.com
ZHENLIANG ZHANG*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, PR China
*
e-mail: zhliang_zhang@163.com
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Abstract

For any x in $[0,1)$ , let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $ . For any positive integers s and t, we study the set

$$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$

and determine its Hausdorff dimension.

Keywords

continued fractionpartial quotientHausdorff dimension

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 404 - 411
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by the Program of Chongqing Municipal Education Commission (Nos. KJQN202100528 and KJQN202000531), Projects from Chongqing Municipal Science and Technology Commission (No. cstc2018jcyjAX0277) and the Foundation of Chongqing Normal University (No. 20XLB030).

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