  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.
