Title: Product-free sets in SL_n(F_q)
Abstract: Given a group $G$, a subset $A\subset G$ is called product-free if for all $a, b\in A$, $ab
otin A$. Following work by Sarnak and Xue, Gowers (2008) proved that in an $r$ - quasirandom group, the density of a product-free set is bounded by $r^{-\frac{1}{3}}$. Finding the correct upper bound remains an open problem in many groups.
In this talk, we present an asymptotic bound for the density of product-free subsets of $SL_n(F_q)$ that is tight up to a sub-exponential factor. Specifically, we show that the density of a product free subset of $SL_n(F_q)$ is at most $q^{O(\sqrt{n})}q^{-\frac{n}{2}}$, improving the Gowers bound of $q^{O(1)}q^{-\frac{n}{3}}$. Our proof employs the concept of character levels introduced by Guralnick, Larsen, and Tiep, alongside hypercontractivity for global functions on $SL_n(F_q)$.
This talk is based on joint work with Noam Lifshitz.