On Shalev's Quantitative Version of Thompson's Conjecture

Abstract

The classical Thompson's conjecture asserts that in any finite nonabelian simple group $G$, there exists a conjugacy class $S$ whose square is equal to the entire group. Shalev (Annals Math., 2009) conjectured that a much stronger quantitative version holds: there exists $\epsilon>0$ such that for any conjugacy class $S$ with $|S| \geq |G|^{1-\epsilon}$, we have $G \setminus \{1\} \subseteq S^2$. While the conjecture turns out to fail in general, we prove a close variant of the conjecture over $\operatorname{PSL}_n(2)$: There exist $\epsilon>0$ and $N \in \mathbb{N}$ such that for all $n\geq N$, every conjugacy class $S\subseteq\operatorname{PSL}_n(2)$ with $|S|\geq|\operatorname{PSL}_n(2)|^{1-\epsilon}$ satisfies $\{x\in\operatorname{PSL}_n(2):\operatorname{supp}(x)\neq1\}\subseteq SS^{-1}.$ On the other hand, we show that for each $\epsilon>0$, there exist conjugacy classes $S \subset \operatorname{PSL}_n(8)$ of size $>|\operatorname{PSL}_n(8))|^{1-\epsilon}$ for which $SS^{-1}$ misses even elements with support size linear in $n$. In addition, for every odd $n \ge 5$, we construct an inverse-closed normal set in $\operatorname{PSL}_n(5)$ with fixed positive density whose square misses a nonidentity element. This answers on the negative a question of Shalev (ECM, 2021) and disproves a conjecture of Pyber (Kourovka notebook, 2022).