A note on the bilinear Bogolyubov theorem: Transverse and bilinear sets
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- by Pierre-Yves Bienvenu, Diego González-Sánchez and Ángel D. Martínez PDF
- Proc. Amer. Math. Soc. 148 (2020), 23-31 Request permission
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.References
- Pierre-Yves Bienvenu and Thái Hoàng Lê, A bilinear Bogolyubov theorem, European J. Combin. 77 (2019), 102–113. MR 3892053, DOI 10.1016/j.ejc.2018.11.003
- N. Bogolioùboff, Sur quelques propriétés arithmétiques des presque-périodes, Ann. Chaire Phys. Math. Kiev 4 (1939), 185–205 (Ukrainian, with French summary). MR 20164
- W. T. Gowers and L. Milićević, A bilinear version of Bogolyubov’s theorem, preprint, available at https://arxiv.org/abs/1712.00248, 2017.
- K. Hosseini and S. Lovett, A bilinear Bogolyubov-Ruzsa lemma with poly-logarithmic bounds, preprint, available at https://arxiv.org/abs/1808.04965, 2018.
- Pierre Samuel, Géométrie projective, Mathématiques. [Mathematics], Presses Universitaires de France, Paris, 1986 (French). MR 850482
- Tom Sanders, On the Bogolyubov-Ruzsa lemma, Anal. PDE 5 (2012), no. 3, 627–655. MR 2994508, DOI 10.2140/apde.2012.5.627
Additional Information
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 23-31
- MSC (2010): Primary 11B30
- DOI: https://doi.org/10.1090/proc/14658
- MathSciNet review: 4042825