A Bilinear Bogolyubov Argument in Abelian Groups

Abstract

The bilinear Bogolyubov argument for ${\mathbb{F}}_p^n$ states that if we start with a dense set $A\subseteq {\mathbb{F}}_p^n\times {\mathbb{F}}_p^n$ and carry out sufficiently many steps where we replace every row or every column of $A$ by the set difference of it with itself, then inside the resulting set we obtain a bilinear variety of codimension bounded in terms of density of $A$. In this paper, we generalize the bilinear Bogolyubov argument to arbitrary finite abelian groups. Namely, if $G$ and $H$ are finite abelian groups and $A\subseteq G\times H$ is a subset of density $\hat{I}´$, then the procedure above applied to $A$ results in a set that contains a bilinear analogue of a Bohr set, with the appropriately defined codimension bounded above by ${\log }^{O(1)}(O(\hat{I}{´}^{-1}))$.

41 pages, final version