The Robust Thompson's Conjecture for Alternating Groups

We show that there exists a constant $\epsilon >0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $\mathrm{\mid }{A}_n{\mathrm{\mid }}^{1-\epsilon }$, 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 proof combines character bounds and combinatorial cancellation techniques with the recent theory of hypercontractivity for functions over symmetric groups.