# American Institute of Mathematical Sciences

May  2018, 38(5): 2375-2393. doi: 10.3934/dcds.2018098

## Hausdorff dimension of certain sets arising in Engel continued fractions

 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

Received  July 2017 Revised  November 2017 Published  March 2018

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.
Citation: Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
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