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

Let G be a finite Abelian group and A subset or equal GxG 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≠0. This is a two-dimensional generalization of Szemeredi's theorem on arithmetic progressions.