A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets

Abstract

A set $P\subset {\mathbb{F}}_p^n\times {\mathbb{F}}_p^n$ is called $\text{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\text{transverse}$ when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set $P\subset {\mathbb{F}}_p^n\times {\mathbb{F}}_p^n$ contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.

10 pages, 2 figures