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 C|G|/|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 > 0 and subsets A, B G of finite simple groups of Lie type, if |B| |B||A|^c for some 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.