Thompson's Conjecture for Large Unitary Groups

We prove that every simple unitary group S of sufficiently large order satisfies Thompson's conjecture, that is, S contains a conjugacy class C such that C² = S. The proof relies on exponential character bounds recently obtained in arXiv:1708.03844 and [LT3], and a refinement that gives a sharp asymptotic for character ratios in Lie-type groups of type A.