Hausdorff dimension of certain sets arising in Engel continued fractions

Lulu Fang; Min Wu

doi:10.3934/dcds.2018098

Article Contents

2018, Volume 38, Issue 5: 2375-2393. Doi: 10.3934/dcds.2018098

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Hausdorff dimension of certain sets arising in Engel continued fractions

Lulu Fang^1, , and
Min Wu^2,

1.
School of Mathematics, Sun Yat-sen University, Guangzhou, GD 510275, China
2.
Department of Mathematics, South China University of Technology, Guangzhou, GD 510640, China

* Corresponding author: Lulu Fang
* Corresponding author: Lulu Fang

Received: July 2017

Revised: November 2017

Published: July 2018

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Abstract

In the present paper, we are concerned with the Hausdorff dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorff dimension of sets

$\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$

are completely determined, where $i.m.$ means infinitely many, $\{b_n(x)\}_{n ≥ 1}$ is the sequence of partial quotients of the Engel continued fraction expansion of $x$ and $\phi$ is a positive function defined on natural numbers.

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Mathematics Subject Classification: Primary: 11K50, 37E05; Secondary: 28A80.

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