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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Uniformly non–improvable Dirichlet set via continued fractions
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by Lingling Huang and Jun Wu PDF
Proc. Amer. Math. Soc. 147 (2019), 4617-4624 Request permission

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
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4617-4624
  • MSC (2010): Primary 11K50, 11J70, 28A80
  • DOI: https://doi.org/10.1090/proc/14587
  • MathSciNet review: 4011499