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

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