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Uniform Diophantine approximation and run-length function in continued fractions

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

Published online by Cambridge University Press:  30 September 2024

BO TAN  and
QING-LONG ZHOU Open the ORCID record for QING-LONG ZHOU [Opens in a new window]
BO TAN
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, PR China (e-mail: tanbo@hust.edu.cn)
QING-LONG ZHOU*
Affiliation:
School of Mathematics and Statistics, Wuhan University of Technology, 430070 Wuhan, PR China
*
e-mail: zhouql@whut.edu.cn
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Abstract

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$
for a class of quadratic irrational numbers $y\in [0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

Keywords

uniform Diophantine approximationcontinued fractionsrun-length function

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 28A80: Fractals 11J83: Metric theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 4 , April 2025 , pp. 1246 - 1280
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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