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 the 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.