Combinatorial Results Implied by Many Zero Divisors in a Group Ring

In a paper of Croot, Lev and Pach and a later paper of Ellenberg and Gijswijt, it was proved 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 \(A\subset G\) 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, a large cyclic group. The aim of this note is to show that the algebraic property corresponding to this difference is the following: in the first case, a group algebra \(\mathbb{F}[G]\) over a 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: the Heisenberg group over \(\mathbb{F}_p\) and the unitriangular group over \(\mathbb{F}_p\). We also show how the method allows us to generalize the results of [14] and [12].