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

Published online by Cambridge University Press:  17 December 2021

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

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.

Type
Research Article
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).

References

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