A bilinear Bogolyubov theorem. (March 2019)

• Record Type: Journal Article
• Authors: Bienvenu, Pierre-Yves; Lê, Thái Hoàng
• Is Part Of: European journal of combinatorics. Volume 77(2019)
• Page Start: 102
• Page End: 113
• Publication Date: 2019-03
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2018.11.003
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set P in a Cartesian square F p n × F p n . More precisely, we perform horizontal and vertical sums and differences on P, that is, operations on the second coordinate when the first one is fixed, or vice versa. The structure we find is the zero set of a family of bilinear forms on a Cartesian product of vector subspaces. The codimensions of the subspaces and the number of bilinear forms involved are bounded by a function c ( δ ) of the density δ = | P | ∕ p 2 n only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog–Szemerédi–Gowers and Freiman–Ruzsa theorems.