On the exponent of convergence of Engel series

Abstract

For $x\in (0,1)$, let $⟨d_1(x),d_2(x),d_3(x),\cdots ⟩$ be the Engel series expansion of $x$. Denote by $\lambda (x)$ the exponent of convergence of the sequence $\{d_n(x)\}$, namely \begin{equation*} λ(x)= \inf\left\{s \geq 0: \sum_{n \geq 1} d^{-s}_n(x)<\infty\right\}. \end{equation*} It follows from Erdős, Rényi and Szüsz (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: \[ \dim_{\rm H} \big\{x \in (0,1): λ(x) =α\big\}=\dim_{\rm H} \big\{x \in (0,1): λ(x) \geqα\big\}= \left\{ \begin{array}{ll} 1-α, & \hbox{$0\le \alpha \le 1$;} 0, & \hbox{$1<\alpha \le \infty$.} \end{array} \right. \]

15 pages

References in corpus (1)

• Multifractal analysis of the convergence exponent in continued fractions