A note on transverse sets and bilinear varieties

TL;DR

This paper provides a direct combinatorial proof that dense transverse sets over finite fields contain bilinear varieties of bounded codimension, improving bounds and avoiding Fourier analysis.

Contribution

It offers a new combinatorial proof for the existence of bilinear varieties in dense transverse sets, improving bounds and simplifying previous methods.

Findings

Dense transverse sets contain bilinear varieties of bounded codimension.

The proof improves bounds and avoids Fourier analysis.

The approach simplifies the understanding of the structure of transverse sets.

Abstract

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns $\{y \in H \colon (x,y) \in A\}$, $x \in G$, are subspaces of $H$. As a corollary of a bilinear version of Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman's theorem and its variants.