Uniform Character Bounds for Finite Classical Groups

For every finite quasisimple group of Lie type G, every irreducible character χ of G, and every element g of G, we give an exponential upper bound for the character ratio |χ(g)|/χ(1) with exponent linear in log|G| |gG|, or, equivalently, in the ratio of the support of g to the rank of G. We give several applications, including a proof of Thompson's conjecture for all sufficiently large simple symplectic groups, orthogonal groups in characteristic $2$, and some other infinite families of orthogonal and unitary groups