The Robust Thompson's Conjecture for Alternating Groups

We show that there exists a constant $ε> 0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-ε}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our p