On the exponent of convergence of Engel series

TL;DR

This paper investigates the topological and fractal properties of the level sets of the exponent of convergence in Engel series expansions, revealing their uncountability, density, and precise Hausdorff dimensions.

Contribution

It characterizes the topological and fractal structure of level sets of the convergence exponent in Engel series, including their Baire category and Hausdorff dimension.

Findings

Each level set is uncountable and dense in (0,1).

Level sets are of the first Baire category for finite exponents.

Hausdorff dimension of level sets is 1−α for 0≤α≤1, and 0 for α>1.

Abstract

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: \[ \dim_{\rm H} \big\{x \in (0,1): \lambda(x) =\alpha\big\}=\dim_{\rm H} \big\{x \in (0,1): \lambda(x) \geq\alpha\big\}= \left\{ \begin{array}{ll} 1-\alpha, & \hbox{$0\leq \alpha\leq1$;} 0, & \hbox{$1<\alpha \leq \infty$.} \end{array} \right. \]