A note on transverse sets and bilinear varieties

This is the authors' abstract. We don't add key points for this paper.

Let G and H be finite-dimensional vector spaces over Fₚ . A subset A ⊆ G × H is said to be transverse if all of its rows ∈ G (x,y) ∈ A\ , y ∈ H , are subspaces of G and all of its columns ∈ H (x,y) ∈ A\ , x ∈ 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.