Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word length of $g$ w.r.t. the generating set $S$.

I would like to know how dense the following set is on the ideal boundary of $G$ (denote by $\partial G$); $$ E_L := \{ \xi_g : l_S(g) \le L \} $$ More precisely, is the set $E_L\quad$ $e^{-\alpha L}$-dense in $\partial G$ for some positive number $\alpha$? If so, for which $\alpha$? I suspect there is some relation between the optimal $\alpha$ and the Hausdorff dimension of $\partial G$.. but not sure how I should investigate further.

Thank you.