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Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2020

A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets

Résumé

A set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ is called $\textit{bilinear}$ when it is the zero set of a family of linear and bilinear forms, and $\textit{transverse}$ when it is stable under vertical and horizontal sums. A theorem of the first author provides a generalization of Bogolyubov's theorem to the bilinear setting. Roughly speaking, it implies that any dense transverse set $P\subset \mathbb{F}_p^n\times\mathbb{F}_p^n$ contains a large bilinear set. In this paper, we elucidate the extent to which a transverse set is forced to be (and not only contain) a bilinear set.

Dates et versions

hal-02080872 , version 1 (27-03-2019)

Identifiants

Citer

Pierre-Yves Bienvenu, Diego González-Sánchez, Ángel D. Martínez. A note on the Bilinear Bogolyubov Theorem: Transverse and bilinear sets. Proceedings of the American Mathematical Society, 2020, 148 (1), pp.23-31. ⟨10.1090/proc/14658⟩. ⟨hal-02080872⟩
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