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 $|A|\ge 2$, there exists $C\log|G|/\log|A|$ 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 $|B| < |G|^{1 - \epsilon}$, then $|A^{\sigma} B| > |B||A|^{c \epsilon}$ 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.