A Bilinear Bogolyubov Argument in Abelian Groups
Discrete analysis (2024)
The bilinear Bogolyubov argument for $\mathbb{F}_p^n$ states that if we start with a dense set $A \subseteq \mathbb{F}_p^n \times \mathbb{F}_p^n$ and carry out sufficiently many steps where we replace every row or every column of $A$ by the set difference of it with itself, then inside the resulting set we obtain a bilinear variety of codimension bounded in terms of density of $A$. In this paper, we generalize the bilinear Bogolyubov argument to arbitrary finite abelian groups. Namely, if $G$ and $H$ are finite abelian groups and $A \subseteq G \times H$ is a subset of density $δ$, then the procedure above applied to $A$ results in a set that contains a bilinear analogue of a Bohr set, with the appropriately defined codimension bounded above by $\log^{O(1)} (O(δ^{-1}))$.
@article{DAS_2024_a1,
author = {L. Mili\'cevi\'c},
title = {A {Bilinear} {Bogolyubov} {Argument} in {Abelian} {Groups}},
journal = {Discrete analysis},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DAS_2024_a1/}
}
L. Milićević. A Bilinear Bogolyubov Argument in Abelian Groups. Discrete analysis (2024). http://geodesic.mathdoc.fr/item/DAS_2024_a1/