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Published online by Cambridge University Press:  11 October 2021

College of Information and Intelligence, Hunan Agricultural University, Changsha 410128, PR China e-mail:
Faculty of Information Technology, Macau University of Science and Technology, Macau, PR China


This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

Research Article
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Communicated by Dzmitry Badziahin

This work was supported by NSFC (Grant Nos. 12001190 and 11871208) and the Science and Technology Development Fund, Macau SAR (no. 0024/2018/A1).


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