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# ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS

Published online by Cambridge University Press:  13 February 2020

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.

## MSC classification

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , October 2020 , pp. 196 - 206
© 2020 Australian Mathematical Publishing Association Inc.

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## Footnotes

This work is supported by NSFC Grant No. 11431007.

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