Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle D^{c}(\tau )=\Big \{x\in [0,1): \limsup \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}\ge \tau \Big \}$    

is of Hausdorff dimension $ \frac {2}{\tau +2}.$ In this note, we study the Hausdorff dimension of the set

  $\displaystyle F(\tau )=\Big \{x\in [0,1): \lim \limits _{n\rightarrow \infty }\frac {\log \big (a_n(x)a_{n+1}(x)\big )}{\log q_n(x)}=\tau \Big \}.$    

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.$

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11K50, 11J70, 28A80

Retrieve articles in all journals with MSC (2010): 11K50, 11J70, 28A80


Additional Information

Lingling Huang
Affiliation: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
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

DOI: https://doi.org/10.1090/proc/14587
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