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 |

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. |

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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