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

Abstract

AbstractFor every integerkthere exists a bound$$B=B(k)$$B=B(k)such that if the characteristic polynomial of$$g\in \textrm{SL}_n(q)$$g∈SLn(q)is the product of$$\le k$$≤kpairwise distinct monic irreducible polynomials over$$\mathbb {F}_q$$Fq, then every elementxof$$\textrm{SL}_n(q)$$SLn(q)of support at leastBis the product of two conjugates ofg. 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$$n=p$$n=pis prime, ifghas order$$\frac{q^p-1}{q-1}$$qp-1q-1, then every non-scalar element$$x \in \textrm{SL}_p(q)$$x∈SLp(q)is the product of two conjugates ofg. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.