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 \begin{document}$\big\{x ∈ [0,1): b_n(x) ≥ \phi (n)~i.m.~n ∈ \mathbb{N}\big\}\ \ \text{and}\ \ \big\{x ∈ [0,1): b_n(x) ≥ \phi(n),\ \forall n ≥ 1\big\}$ \end{document} 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.
Highlights
Given a real number, there are various ways to represent it as an expansion of digits or partial quotients, such as continued fractions and series expansions
The Hausdorff dimension of sets x ∈ [0, 1) : bn(x) ≥ φ(n) i.m. n ∈ N and x ∈ [0, 1) : bn(x) ≥ φ(n), ∀n ≥ 1 are completely determined, where i.m. means infinitely many, {bn(x)}n≥1 is the sequence of partial quotients of the Engel continued fraction expansion of x and φ is a positive function defined on natural numbers
The regular continued fraction expansion of a real number can be induced by the RCF-map T : [0, 1) → [0, 1) given by
Summary
There are various ways to represent it as an expansion of digits or partial quotients, such as continued fractions (see Khintchine [17]) and series expansions (see Galambos [9] and Schweiger [23]). The Hausdorff dimension of sets x ∈ [0, 1) : bn(x) ≥ φ(n) i.m. n ∈ N and x ∈ [0, 1) : bn(x) ≥ φ(n), ∀n ≥ 1 are completely determined, where i.m. means infinitely many, {bn(x)}n≥1 is the sequence of partial quotients of the Engel continued fraction expansion of x and φ is a positive function defined on natural numbers.
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