Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as \emph{normal} sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and $A$ is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon}}$, then $A^2 \supseteq A_n$. This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$. Our proof strategy combines two types of techniques. The first is `traditional' techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck--Shalev, Larsen--Shalev, and more recently, Larsen--Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.