For $x\in (0,1)$, let $\langle d_1(x),d_2(x),d_3(x),\cdots \rangle$ be the Engel series expansion of $x$. Denote by $\lambda(x)$ the exponent of convergence of the sequence $\{ d_n(x)\}$, namely \begin{ equation*} \lambda(x)= \inf\left\{ s \geq 0: \sum_{ n \geq 1} d^{ -s}_n(x)<\infty\right\}. \end{ equation*} It follows from Erd\H{ o}s, R\'{ e}nyi and Sz\"{ u}sz (1958) that $\lambda(x) =0$ for Lebesgue almost all $x\in (0,1)$. This paper is concerned with the topological and fractal properties of the level set $\{ x\in (0,1): \lambda(x) =\alpha\}$ for $\alpha \in [0,\infty]$. For the topological properties, it is proved that each level set is uncountable and dense in $(0,1)$. Furthermore, the level set is of the first Baire category for $\alpha\in [0,\infty)$ but residual for $\alpha =\infty$. For the fractal properties, we prove that the Hausdorff dimension of the level set is as follows: \[ \dim_{ \rm H} \big\{ x \in (0,1): \lambda(x) =\alpha\big\}=\dim_{ \rm H} \big\{ x \in (0,1): \lambda(x) \geq\alpha\big\}= \left\{ \begin{ array}{ ll} 1-\alpha, & \hbox{ $0\leq \alpha\leq1$;} 0, & \hbox{ $1<\alpha \leq \infty$.} \end{ array} \right. \]
Categories: math.NT math.CT