  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
$\mathbb{F}_p^n\times\mathbb{F}_p^n$. 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 $c(\delta)$ of the density $\delta=\lvert
P\rvert/p^{2n}$ only. The proof uses various tools of additive combinatorics,
such as the (linear) Bogolyubov theorem, the density increment method, as well
as the Balog-Szemer\'edi-Gowers and Freiman-Ruzsa theorems.
