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

Let G be a finite Abelian group and A subset of G x G a set of cardinality at least vertical bar G vertical bar(2)/(log log vertical bar G vertical bar)(c), where c > 0 is air absolute constant. We prove that A contains a triple {(k,m), (k + d, m), (k,m + d)} with d not equal 0. This is a two-dimensional generalization of Szemeredi's theorem on arithmetic progressions.