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ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  13 February 2020

TENG SONG Open the ORCID record for TENG SONG [Opens in a new window]  and
QINGLONG ZHOU Open the ORCID record for QINGLONG ZHOU [Opens in a new window]
TENG SONG
Affiliation:
School of Mathematics and Statistics,Huazhong University of Science and Technology, Wuhan, PR China email [email protected]
QINGLONG ZHOU*
Affiliation:
School of Mathematics and Statistics,Huazhong University of Science and Technology, Wuhan, PR China email [email protected]
*
✉[email protected]
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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.

Keywords

continued fractionlongest block functionHausdorff dimension

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 11J83: Metric theory 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 196 - 206
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by NSFC Grant No. 11431007.

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