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

Abstract

For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of gin textrm{SL}_n(q) is the product of le k pairwise distinct monic irreducible polynomials over mathbb {F}_q, then every element x of textrm{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 frac{q^p-1}{q-1}, then every non-scalar element x in textrm{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.