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For every integer $k$ there exists a bound $B=B(k)$ such that if thecharacteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of$\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$,then every element $x$ of $\operatorname{SL}_n(q)$ of support at least $B$ isthe product of two conjugates of $g$. We prove this and analogous results forthe other classical groups over finite fields; in the orthogonal and symplecticcases, the result is slightly weaker. With finitely many exceptions $(p,q)$, inthe special case that $n=p$ is prime, if $g$ has order $\frac{q^p-1}{q-1}$,then every non-scalar element $x \in \operatorname{SL}_p(q)$ is the product oftwo conjugates of $g$. The proofs use the Frobenius formula together with upperbounds for values of unipotent and quadratic unipotent characters in finiteclassical groups.

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