Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

For every integer k there exists a bound B= B(k) such that if the characteristic polynomial of g∈ SL _n(q) is the product of ≤ k pairwise distinct monic irreducible polynomials over F_q, then every element x of SL _n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n= p is prime, if g has order qp-1q-1, then every non-scalar element x∈ SL _p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.