A bilinear version of Bogolyubov's theorem

A theorem of Bogolyubov states that for every dense set $A$ in ${\mathbb{Z}}_N$ we may find a large Bohr set inside $A+A-A-A$. In this note, motivated by the work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a bilinear variant of this result in vector spaces over finite fields. Namely, if we start with a dense set $A\subset {\mathbb{F}}_p^n\times {\mathbb{F}}_p^n$ and then take rows (respectively columns) of $A$ and change each row (respectively column) to the set difference of it with itself, repeating this procedure several times, we obtain a bilinear analogue of a Bohr set inside the resulting set, namely the zero set of a biaffine map from $$\mathbb{F}$^n_p \times $\mathbb{F}$^n_p$toa$$\mathbb{F}$_p$-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and L^e.