关于Engel级数的收敛指数

对于$x\in (0,1)$,令$\langle d_1(x),d_2(x),d_3(x),\cdots \rangle$为$x$的Engel级数展开。记$\lambda(x)$为序列$\{d_n(x)\}$的收敛指数,即\begin{equation*} \lambda(x)= \inf\left\{s \geq 0: \sum_{n \geq 1} d^{-s}_n(x)<\infty\right\}. \end{equation*}由Erd\H{o}s、R\'{e}nyi和Sz"{u}sz(1958)可知,对Lebesgue几乎处处的$x\in (0,1)$有$\lambda(x) =0$。本文关注水平集$\{x\in (0,1): \lambda(x) =\alpha\}$(其中$\alpha \in [0,\infty]$)的拓扑与分形性质。对于拓扑性质,证明了每个水平集在$(0,1)$中不可数且稠密。进一步,对$\alpha\in [0,\infty)$水平集为第一Baire纲集,而对$\alpha =\infty$为剩余集。对于分形性质,我们证明水平集的Hausdorff维数如下:$$\dim_{\rm H} \big\{x \in (0,1): \lambda(x) =\alpha\big\}=\dim_{\rm H} \big\{x \in (0,1): \lambda(x) \geq\alpha\big\}= \left\{ \begin{array}{ll} 1-\alpha, & \hbox{$0\leq \alpha\leq1$;} 0, & \hbox{$1<\alpha \leq \infty$.} \end{array} \right.$$