Improved covering results for conjugacy classes of symmetric groups via hypercontractivity (Q6633890)

For an element \(\sigma\) of a finite group \(G\), let \(\sigma^{G}\) be its conjugacy class in \(G\). Let \(S_n\) be the symmetric group of degree \(n\) and let \(A_n\) be the alternating group of degree \(n\). It is proved in the paper that for any \(\epsilon > 0\), there exists an integer \(n_0\), such that for any \(n > n_0\) and for any \(\sigma \in S_n\) that has at most \(n^{2/5 - \epsilon}\) cycles, we have \((\sigma^{S_n})^{2} = A_n\).\N\NLet \(A\) be a subset of a finite group \(G\). Write \(\mu(A)\) for \(|A|/|G|\). If \(A\) is a union of conjugacy classes of \(G\), then \(A\) is called a normal set. It is also proved in the paper that for any \(\epsilon > 0\), there exists an integer \(n_0\), such that for any \(n > n_0\) and for any normal subset \(A\) of \(S_n\) with \(\mu(A) \geq e^{-n^{2/5 - \epsilon}}\), the group \(A_n\) is contained in \(A^{2}\).\N\NThese two results improve upon [\textit{M. Larsen} and \textit{A. Shalev}, Invent. Math. 174, No. 3, 645--687 (2008; Zbl 1166.20009), Theorem 1.10 and Theorem 1.20] respectively from a breakthrough paper of Larsen and Shalev with \(2/5\) in the exponents replacing their \(1/4\).\N\NThere is a version of the latter theorem of the paper under review for normal subsets \(A\) of \(A_n\) (rather than \(S_n\)) where the conclusion is that \(A_{n} \setminus \{ 1 \}\) is contained in \(A^2\). This is a strengthening of a result of \textit{M. J. Larsen} and \textit{P. H. Tiep} [``Squares of Conjugacy Classes in Alternating Groups'', Preprint, \url{arXiv:2305.04806}] and of [``Bounds for characters of the symmetric group: a hypercontractive approach'', Preprint, \url{arXiv:2308.08694}, Corollary 2.11] by \textit{N. Lifshitz} and \textit{A. Marmor}.\N\NA motivating conjecture in the area is due to Thompson: every non-abelian finite simple group \(G\) contains a conjugacy class \(A\) such that \(A^{2} = G\).\N\NA subset of the vertices of a graph is called independent if it does not contain an edge. A Cayley graph \(\mathrm{Cay}(G,A)\) of a finite group \(G\) with generating set \(A\) is called normal if \(A\) is normal in \(G\). The second set of results in the paper under review concerns independent sets in normal Cayley graphs. The following is proved. For any \(\epsilon > 0\), there exist \(\delta\), \(n_0\), such that the following holds for any natural number \(t\) and \(n > n_{0}+t\). Let \(\sigma \in A_n\) be a permutation with \(t\) fixed points. Then the largest independent set in \(\mathrm{Cay}(A_{n},\sigma^{S_n})\) has density at most \(\max (e^{-(n-t)^{1/3 - \epsilon}}, (n-t)^{-\delta t})\).\N\NThe paper uses both algebraic and analytic methods. Algebraic tools include character bounds and asymptotics for the Witten zeta function. The authors apply an interesting new approach, the analysis of Boolean functions, in particular, a sharp hypercontractivity theorem in the symmetric group recently obtained by \textit{P. Keevash} and \textit{N. Lifshitz} [``Sharp hypercontractivity for symmetric groups and its applications'', Preprint, \url{arXiv:2307.15030}, Theorem 4.1]. The paper is nicely written and well structured.