Character Estimates of Adjoint Simple Lie Groups

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$, depending on $G$ but not $C$, such that $e$ lies in the interior of $C^n$ for all $n\geq N$. We then prove that a disk $D\subset\mathbb{C}$ of radius less than 1, contained in the unit disk $D_1$ and tangent to $D_1$ at $z=1$, contains the image of every normalized character $\chi(e)^{-1}\chi$ of $G$.