Uniform character bounds for finite classical groups

For every finite quasisimple group of Lie type G, every irreducible character of G, and every element g of G, we give an exponential upper bound for the character ratio | (g) |/ (1) with exponent linear in log|₆| |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.