turkmath.org

Bilinear Bogolyubov argument states that if we start with a dense set $A$ of a product $U\times V$ of finite vector spaces 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 the density of $A$. In this talk, we focus on a generalization of this theorem to arbitrary finite abelian groups. Along the way we also discuss analogues of a few linear-algebraic facts as well as quasirandomness of analogues of bilinear varieties in the setting of finite abelian groups.