On subsets of finite abelian groups with no 3-term arithmetic progressions

Let G be a finite abelian group of odd order and let D(G) denote the maximal cardinality of a subset A ⊂ G which does not contain a 3-term arithmetic progression. It is shown that D(Zk1 ⊕ ⋯ ⊕ Zkn) ⩽ 2((k1 ⋯ kn/n). Together with results of Szemerédi and Heath-Brown it implies that there exists a β > 0 such that D(G) = O(∥G∥/(log ∥G∥)β) for all G.