A note on transverse sets and bilinear varieties

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 $\{x\in G\text{\hspace{0.17em}⁣}:(x,y)\in A\}$, $y\in H$, are subspaces of $G$ and all of its columns $\{y\in H\text{\hspace{0.17em}⁣}:(x,y)\in A\}$, $x\in G$, 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.