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

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Contribution

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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 $\vert \omega(G)\vert >c\frac{\vert G\vert }{n}$ whenether $\vert G\vert >N$.