Skip to main content Accessibility help
×
Home
Hostname: page-component-5cfd469876-7frv5 Total loading time: 1.269 Render date: 2021-06-24T20:07:12.025Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS

Published online by Cambridge University Press:  13 February 2020

TENG SONG
Affiliation:
School of Mathematics and Statistics,Huazhong University of Science and Technology, Wuhan, PR China email teng_song@hust.edu.cn
QINGLONG ZHOU
Affiliation:
School of Mathematics and Statistics,Huazhong University of Science and Technology, Wuhan, PR China email qinglong_zhou@hust.edu.cn
Corresponding

Abstract

For an irrational number $x\in [0,1)$ , let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$ . Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$ , for $n\geq 1$ , the $n$ th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$ , which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$ . We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below.

Footnotes

This work is supported by NSFC Grant No. 11431007.

References

Bakhtawar, A., Bos, P. and Hussain, M., ‘The sets of Dirichlet non-improvable numbers vs well-approximable numbers’, Preprint, 2018, arXiv:1806.00618.CrossRefGoogle Scholar
Bugeaud, Y. and Wang, B. W., ‘Distribution of full cylinders and the Diophantine properties of the orbits in 𝛽-expansions’, J. Fractal Geom. 1(2) (2014), 221241.CrossRefGoogle Scholar
Chang, J. H. and Chen, H. B., ‘Slow increasing functions and the largest partial quotients in continued fraction expansions’, Math. Proc. Cambridge Philos. Soc. 164(1) (2018), 114.CrossRefGoogle Scholar
Erdős, P. and Rényi, A., ‘On a new law of large numbers’, J. Analyse Math. 23 (1970), 103111.CrossRefGoogle Scholar
Falconer, K. J., Fractal Geometry. Mathematical Foundations and Applications, 3rd edn (Wiley, Chichester, 2014).Google Scholar
Ge, Y. H. and , F., ‘A note on inhomogeneous Diophantine approximation in beta-dynamical system’, Bull. Aust. Math. Soc. 91(1) (2015), 3440.CrossRefGoogle Scholar
Good, I. J., ‘The fractional dimensional theory of continued fractions’, Proc. Cambridge Philos. Soc. 37 (1941), 199228.CrossRefGoogle Scholar
Huang, L. L. and Wu, J., ‘Uniformly non-improvable Dirichlet set via continued fractions’, Proc. Amer. Math. Soc. 147(11) (2019), 46174624.CrossRefGoogle Scholar
Hussain, M., Kleinbock, D., Wadleigh, N. and Wang, B. W., ‘Hausdorff measure of sets of Dirichlet non-improvable numbers’, Mathematika 64(2) (2018), 502518.CrossRefGoogle Scholar
Jarník, V., ‘Zur metrischen Theorie der diophantischen Approximationen’, Prace Mat. Fiz. 36 (1929), 91106.Google Scholar
Khintchine, A. Ya., Continued Fractions (Noordhoff, Groningen, 1963), translated by Peter Wynn.Google Scholar
Liu, J., , M. Y. and Zhang, Z. L., ‘On the exceptional sets in Erdős–Rényi limit theorem of 𝛽-expansion’, Int. J. Number Theory 14(7) (2018), 19191934.CrossRefGoogle Scholar
Ma, J. H., Wen, S. Y. and Wen, Z. Y., ‘Egoroff’s theorem and maximal run length’, Monatsh. Math. 151(4) (2007), 287292.CrossRefGoogle Scholar
Peng, L., Tan, B. and Wang, B. W., ‘Quantitative Poincaré recurrence in continued fraction dynamical system’, Sci. China Math. 55(1) (2012), 131140.CrossRefGoogle Scholar
Philipp, W., ‘Some metrical theorems in number theory’, Pacific J. Math. 20 (1967), 109127.CrossRefGoogle Scholar
Révész, P., Random Walk in Random and Non-Random Environments, 2nd edn (World Scientific, Hackensack, NJ, 2005).CrossRefGoogle Scholar
Sun, Y. and Xu, J., ‘A remark on exceptional sets in Erdős–Rényi limit theorem’, Monatsh. Math. 184(2) (2017), 291296.CrossRefGoogle Scholar
Tan, B. and Zhou, Q. L., ‘The relative growth rate for partial quotients in continued fractions’, J. Math. Anal. Appl. 478(1) (2019), 229235.CrossRefGoogle Scholar
Tong, X., Yu, Y. L. and Zhao, Y. F., ‘On the maximal length of consecutive zero digits of 𝛽-expansions’, Int. J. Number Theory 12(3) (2016), 625633.CrossRefGoogle Scholar
Wang, B. W. and Wu, J., ‘On the maximal run-length function in continued fractions’, Ann. Univ. Sci. Budapest. Sect. Comput. 34 (2011), 247268.Google Scholar
Wu, J., ‘A remark on the growth of the denominators of convergents’, Monatsh. Math. 147(3) (2006), 259264.CrossRefGoogle Scholar
Zhou, Q. L., ‘Dimensions of recurrent sets in 𝛽-symbolic dynamics’, J. Math. Anal. Appl. 472(2) (2019), 17621776.CrossRefGoogle Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *