Character estimates for finite classical groups and the asymptotic Thompson Conjecture

If $G$ is a finite classical group, linear or unitary in any characteristic, and orthogonal in odd characteristic, we give an approximate formula for $\chi(g)$ in which the error term is much smaller than the estimate, when $g\in G$ is an element with large centralizer and $\chi\in \mathrm{Irr}(G)$ is an irreducible character of low degree. As an application, we prove Thompson's conjecture for all sufficiently large finite simple groups: each such group contains a conjugacy class whose square is the whole group.