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|≥ 2, there exists Clog|G|/log|A| conjugates of A whose product is G. This paper is a companion to {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 ε > 0 and subsets A, B ⊆ G of finite simple groups of Lie type, if |B| < |G|1 - ε, then |Aσ B| > |B||A|c ε for some σ ∈ 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.