Authors: Gowers, Tim
Milićević, Luka 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: A bilinear version of Bogolyubov’s theorem
Journal: Proceedings of the American Mathematical Society
Volume: 148
Issue: 11
First page: 4695
Last page: 4704
Issue Date: 1-Nov-2020
Rank: M22
ISSN: 0002-9939
DOI: 10.1090/proc/15129
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.
Publisher: American Mathematical Society
Project: Development of new information and communication technologies, based on advanced mathematical methods, with applications in medicine, telecommunications, power systems, protection of national heritage and education 
Representations of logical structures and formal languages and their application in computing 

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