Polynomial Bogolyubov for special linear groups via tensor rank

Abstract

We prove a polynomial Bogolyubov type lemma for the special linear group over finite fields. Specifically, we show that there exists an absolute constant $C>0,$ such that if $A$ is a density $\alpha$ subset of the special linear group, then the set $AA^{-1}AA^{-1}$ contains a subgroup $H$ of density $\alpha^C$. Moreover, this subgroup is isomorphic to a special linear group of a smaller rank. We also show that if $A$ is an approximate subgroups then it can be covered by the union of few cosets of $H$. Our proof makes use of the Gurevich--Howe notion of tensor rank, and of a strengthened Bonami type Lemma for global functions on the bilinear scheme. We also present applications to spectral bounds for global convolution operators, global product free sets, and covering numbers corresponding to global sets.