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ON THE SHORTEST DISTANCE FUNCTION IN CONTINUED FRACTIONS

Part of: Elementary number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  23 June 2023

SAISAI SHI Open the ORCID record for SAISAI SHI [Opens in a new window]  and
QINGLONG ZHOU Open the ORCID record for QINGLONG ZHOU [Opens in a new window]
SAISAI SHI
Affiliation:
Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, P. R. China e-mail: saisai_shi@126.com
QINGLONG ZHOU*
Affiliation:
School of Science, Wuhan University of Technology, Wuhan 430070, P. R. China
*
e-mail: zhouql@whut.edu.cn
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Abstract

Let $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$. Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function

$$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$

which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function.

Keywords

shortest distance functionHausdorff dimensioncontinued fraction

MSC classification

Primary: 11A55: Continued fractions
Secondary: 11K50: Metric theory of continued fractions 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 109 , Issue 2 , April 2024 , pp. 186 - 195
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by National Natural Science Foundation of China (NSFC), No. 12201476.

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