A bilinear version of Bogolyubov’s theorem

Abstract

A theorem of Bogolyubov states that for every dense set A in ZN 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 F^np F^np, 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 F^np F^np to an Fp-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.