Hausdorff dimensions of some exceptional sets in Engel expansions
In this paper, it is noted that the condition $\lim_{n\to\infty}{\frac{\log {d_n(x)}}{n}}=1$ holds for almost all numbers $x\in (0,1]$ represented by the Engel expansion \[ x=\sum^{\infty} _{n=1}{\frac{1}{d_1(x)d_2(x)\cdots d_n(x)}}, \] where $(d_n(x))$ is a sequence of positive integers satisfying the conditions $d_1(x)\ge 2$ and $d_{n+1}(x)\ge d_{n}(x)$, where $n=1,2,\dots$. \par In the present article, the main attention is given to the Hausdorff dimension of the set \[ E(\phi)=\left\{x\in(0,1]: \lim_{n\to\infty}{ \frac{\log {d_n(x)}}{\phi(n)}}=1\right\}, \] where $x$ is represented by the Engel expansion and $\phi(n): \mathbb N \to \mathbb R^{+}$ is a function satisfying $\phi(n+1)-\phi(n)\to\infty$ as $n\to\infty$. Also, some partial cases of $E(\phi)$ are considered: \begin{itemize} \item the case when $\phi(n)=\gamma n^{\beta}$, where $\gamma>0$ and $\beta>1$; \item the case when $\phi(n)=\eta\tau^{n}$, where $\eta>0$ and $ \tau>1$. \end{itemize} The main statements on the Hausdorff dimension of these sets are proved. Certain well-known properties of the representation of real numbers by an Engel expansion, are described. A formula for computing the Hausdorff dimension for a class of Cantor sets related to Engel expansions is given.
scientific article; zbMATH DE number 6820896