A note on transverse sets and bilinear varieties

Abstract

Let G and H be finite-dimensional vector spaces over F<inf>p</inf>. A subset A ⊆ G × H is said to be transverse if all of its rows {x ∈ G: (x, y) ∈ A}, y ∈ H, are subspaces of G and all of its columns {y ∈ 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.