A bilinear Bogolyubov theorem

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The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set P in a Cartesian square Fₚⁿₚⁿ. More precisely, we perform horizontal and vertical sums and differences on P, that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function c(δ) of the density δ= P/p²ⁿ only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog-Szemerédi-Gowers and Freiman-Ruzsa theorems.