On Shalev's conjecture for type $A_n$ and ${}^2{A}_n$

Abstract

In the paper we consider images of finite simple projective special linear and unitary groups under power words. In particular, we show that if $G\simeq \PSL_n^\varepsilon (q)$, then for every power words of type $x^M$ there exist constant $c$ and $N$ such that $|\omega (G)|>c\frac{|G|}{n}$ whenether $|G|>N$.