2017/12/01 by Gowers, W. T., Milićević, L. · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1712.00248
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 the work on a quantitative inverse theorem for the Gowers U4 norm, we prove a bilinear variant of this result in vector spaces over finite fields. Namely, if we start with a dense set A ⊂ \mathbbFnp × \mathbbFnp 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 \mathbbFnp × \mathbbFnp to a \mathbbFp-vector space of bounded dimension. An almost identical result was proved independently by Bienvenu and Lê.