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A note on the bilinear Bogolyubov theorem: Transverse and bilinear sets


Authors: Pierre-Yves Bienvenu, Diego González-Sánchez and Ángel D. Martínez
Journal: Proc. Amer. Math. Soc. 148 (2020), 23-31
MSC (2010): Primary 11B30
DOI: https://doi.org/10.1090/proc/14658
Published electronically: July 8, 2019
MathSciNet review: 4042825
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Abstract: A set $P\subset \mathbb {F}_p^n\times \mathbb {F}_p^n$ is called bilinear when it is the zero set of a family of linear and bilinear forms and 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.


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Pierre-Yves Bienvenu
Affiliation: Institut Camille-Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
MR Author ID: 1214852
Email: pbienvenu@math.univ-lyon1.fr

Diego González-Sánchez
Affiliation: Departamento de Matemáticas, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: diego.gonzalezs@uam.es

Ángel D. Martínez
Affiliation: Institute for Advanced Study, Fuld Hall 412, 1 Einstein Drive, Princeton, New Jersey 08540
Email: amartinez@ias.edu

Received by editor(s): November 24, 2018
Received by editor(s) in revised form: April 8, 2019
Published electronically: July 8, 2019
Additional Notes: The first named author was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon within the program “Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR)
The second- and third-named authors were partially supported by MTM2014-56350-P project of the MCINN (Spain).
This material is based upon work supported by the National Science Foundation under grant No. DMS-1638352
Communicated by: Patricia Hersh
Article copyright: © Copyright 2019 American Mathematical Society