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arXiv · 1711.05349

A bilinear Bogolyubov theorem

Pierre-Yves Bienvenu·Thái Hoàng Lê
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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 $\mathbb{F}_p^n\times\mathbb{F}_p^n$. 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 $c(\delta)$ of the density $\delta=\lvert P\rvert/p^{2n}$ only. The proof uses various tools of additive combinatorics, such as the (linear) Bogolyubov theorem, the density increment method, as well as the Balog-Szemer\'edi-Gowers and Freiman-Ruzsa theorems.

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Pierre-Yves Bienvenu, Thái Hoàng Lê. 2017-11-14. A bilinear Bogolyubov theorem. https://arxiv.org/abs/1711.05349

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