Completing the proof of the Liebeck--Nikolov--Shalev conjecture

Liebeck, Nikolov, and Shalev conjectured the existence of an absolute constant $C>0$, such that for every subset $A$ of a finite simple group $G$ with $\mathrm{\mid }A\mathrm{\mid }\ge 2$, there exists $C\log \mathrm{\mid }G\mathrm{\mid }\mathrm{/}\log \mathrm{\mid }A\mathrm{\mid }$ conjugates of $A$ whose product is $G$. This paper is a companion to \cite{GLPS}, and together they prove the conjecture. To prove the conjecture, we establish the following skew-product theorem. We show that there exists $c>0$ such that for all $\epsilon >0$ and subsets $A,B\subseteq G$ of finite simple groups of Lie type, if $\mathrm{\mid }B\mathrm{\mid }<\mathrm{\mid }G{\mathrm{\mid }}^{1-\epsilon }$, then $\mathrm{\mid }{A}^{\sigma }B\mathrm{\mid }>\mathrm{\mid }B\mathrm{\mid }\mathrm{\mid }A{\mathrm{\mid }}^{c\epsilon }$ for some $\sigma \in G$. This result, along with its more involved analogue for alternating groups, constitutes the main contribution of this paper. Our proof leverages deep results from character theory alongside the probabilistic method.