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Uniformly non–improvable Dirichlet set via continued fractions


Authors: Lingling Huang and Jun Wu
Journal: Proc. Amer. Math. Soc. 147 (2019), 4617-4624
MSC (2010): Primary 11K50, 11J70, 28A80
DOI: https://doi.org/10.1090/proc/14587
Published electronically: July 9, 2019
MathSciNet review: 4011499
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Abstract: Let $[a_1(x),a_2(x),\ldots ]$ be the continued fraction expansion of $x\in [0,1)$ and let $q_n(x)$ be the denominator of the $n$th convergent. Recently, Hussain-Kleinbock-Wadleigh-Wang (2018) showed that for any $\tau \ge 0,$ the set \begin{equation*} D^{c}(\tau )=\Big \{x\in [0,1): \limsup \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}\ge \tau \Big \} \end{equation*} is of Hausdorff dimension $\frac {2}{\tau +2}.$ In this note, we study the Hausdorff dimension of the set \begin{align*} &F(\tau )=\Big \{x\in [0,1): \lim \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}=\tau \Big \}. \end{align*} It is proved that the set $F(\tau )$ has Hausdorff dimension $1$ or $\frac {2}{\tau +\sqrt {\tau ^2+4}+2}$ according as $\tau =0$ or $\tau >0.$


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Additional Information

Lingling Huang
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
MR Author ID: 1248934
Email: huanglingling@hust.edu.cn

Jun Wu
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
Email: jun.wu@hust.edu.cn

Keywords: Continued fraction, partial quotients, Hausdorff dimension.
Received by editor(s): July 30, 2018
Received by editor(s) in revised form: February 8, 2019
Published electronically: July 9, 2019
Additional Notes: This work was partially supported by NSFC 11831007
The second author is the corresponding author
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society