On the exponent of convergence of Engel series

For $xin (0,1)$, 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)= infleft{s geq 0: sum_{n geq 1} d^{-s}_n(x)<inftyright}. end{equation*} It follows from ErdH{o}s, R{e}nyi and Sz{u}sz (1958) that $lambda(x) =0$ for Lebesgue almost all $xin (0,1)$. This paper is concerned with the topological and fractal properties of the level set ${xin (0,1): lambda(x) =alpha}$ 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 $alphain [0,infty)$ 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) =alphabig}=dim_{rm H} big{x in (0,1): lambda(x) geqalphabig}= left{ begin{array}{ll} 1-alpha, & hbox{$0leq alphaleq1$;} 0, & hbox{$1<alpha leq infty$.} end{array} right. ]