Combinatorial results implied by many zero divisors in a group ring

Abstract: It has been recently proved (by Croot, Lev and Pach and the subsequent work by Ellenberg and Gijswijt) that for a group $G=G_0^n$, where $G_0\ne {1,-1}^m$ is a fixed finite Abelian group and $n$ is large, any subset $A$ without 3-progressions (triples $x,y,z$ of different elements with $xy=z^2$) contains at most $|G|^{1-c}$ elements, where $c>0$ is a constant depending only on $G_0$. This is known to be false when $G$ is, say, large cyclic group. The aim of this note is to show that algebraic property which corresponds to this difference is the following: in the first case a group algebra $\mathbb{F}[G]$ over suitable field $\mathbb{F}$ contains a subspace $X$ with codimension at most $|X|^{1-c}$ such that $X^3=0$. We discuss which bounds are obtained for finite Abelian $p$-groups and for some matrix $p$-groups: Heisenberg group over $\mathbb{F}_p$ and the unitriangular group over $\mathbb{F}_p$. Also we show how the method works for further generalizations by Kleinberg--Sawin--Speyer and Ellenberg.