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Hausdorff dimension of certain sets arising in Engel continued fractions

  • * Corresponding author: Lulu Fang

    * Corresponding author: Lulu Fang 
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  • 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.

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


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