  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 $A \subset \mathbb{F}^n_p \times \mathbb{F}^n_p$ 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 $\mathbb{F}^n_p \times
\mathbb{F}^n_p$ to a $\mathbb{F}_p$-vector space of bounded dimension. An
almost identical result was proved independently by Bienvenu and L\^e.
