A bilinear Bogolyubov theorem

TL;DR

This paper proves a structure theorem for iterated sumsets in finite fields, showing they contain zero sets of bilinear forms on subspaces with bounds depending only on the set density.

Contribution

It establishes a bilinear Bogolyubov theorem, revealing a structured subset within sumsets in finite fields, with bounds depending solely on the density.

Findings

Identifies a bilinear structure in iterated sumsets.

Bounds the codimensions of subspaces and number of bilinear forms by a function of density.

Uses additive combinatorics tools like Bogolyubov, Balog-Szemerédi-Gowers, and Freiman-Ruzsa theorems.

Abstract

The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums and differences on $P$, that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function $c(\delta)$ of the density $\delta=\lvert P\rvert/p^{2n}$ only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog-Szemer\'edi-Gowers and Freiman-Ruzsa theorems.