A note on transverse sets and bilinear varieties

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.