A Bilinear Bogolyubov Argument in Abelian Groups

TL;DR

This paper extends the bilinear Bogolyubov argument from finite fields to arbitrary finite abelian groups, showing that dense subsets lead to structured bilinear varieties with bounded codimension.

Contribution

It generalizes the bilinear Bogolyubov argument to all finite abelian groups, establishing bounds on the resulting bilinear structures.

Findings

The procedure yields a bilinear variety of bounded codimension.

The codimension bound is logarithmic in the inverse of the density.

The result applies to any finite abelian groups, not just vector spaces.

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 $\delta$, 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 $\log^{O(1)} (O(\delta^{-1}))$.