On a two-dimensional analogue of Szemerédi's theorem in Abelian groups

Let $G$ be a finite Abelian group and $A\subseteq G\times G$ a set of cardinality at least $|G|^2/(\log\log|G|)^c$, where $c>0$ is an absolute constant. We prove that $A$ contains a triple $\{(k,m),(k+d,m),(k,m+d)\}$ with $d\neq0$. This is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.