A bilinear version of Bogolyubov's theorem

TL;DR

This paper extends Bogolyubov's theorem to a bilinear setting over finite fields, showing that dense sets contain structured bilinear objects after iterative difference operations, with implications for inverse theorems in additive combinatorics.

Contribution

It introduces a bilinear analogue of Bogolyubov's theorem in finite field vector spaces, connecting dense sets to zero sets of biaffine maps, advancing understanding of additive structure.

Findings

Establishes a bilinear Bogolyubov-type theorem in finite fields.

Shows the existence of structured bilinear sets within dense subsets after iterative differences.

Provides a new tool for inverse theorems related to Gowers norms.

Abstract

A theorem of Bogolyubov states that for every dense set $A$ in $\mathbb{Z}_N$ we may find a large Bohr set inside $A+A-A-A$. In this note, motivated by the work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a bilinear variant of this result in vector spaces over finite fields. Namely, if we start with a dense set $A \subset \mathbb{F}^n_p \times \mathbb{F}^n_p$ and then take rows (respectively columns) of $A$ and change each row (respectively column) to the set difference of it with itself, repeating this procedure several times, we obtain a bilinear analogue of a Bohr set inside the resulting set, namely the zero set of a biaffine map from $\mathbb{F}^n_p \times \mathbb{F}^n_p$ to a $\mathbb{F}_p$-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and L\^e.