A bilinear version of Bogolyubov’s theorem

Abstract:

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 work on a quantitative inverse theorem for the Gowers $U^4$ norm, we prove a bilinear variant of this result for vector spaces over finite fields. Given a subset $A \subset \mathbb {F}^n_p \times \mathbb {F}^n_p$, we consider two operations: one of them replaces each row of $A$ by the set difference of it with itself, and the other does the same for columns. We prove that if $A$ has positive density and these operations are repeated several times, then the resulting set contains a bilinear analogue of a Bohr set, namely the zero set of a biaffine map from $\mathbb {F}^n_p \times \mathbb {F}^n_p$ to an $\mathbb {F}_p$-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.