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ON THE SHORTEST DISTANCE FUNCTION IN CONTINUED FRACTIONS

Published online by Cambridge University Press:  23 June 2023

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

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.

Type
Research Article
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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