Abstract
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( Z k 1 ⊕ ⋯ ⊕ Z k n ) ⩽ 2(( k 1 ⋯ k n / 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.
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