Asymptotics of the powers in finite reductive groups

Abstract

Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$.