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 m aximal cardinality of a subset A subset of G which does not contain a 3-term arithmetic progression. It is shown that D(Z(k1) + ... + Z(kn)) less than or equal to 2((k(1) ... k(n))/n). Together with results of Szemeredi and Heath-Brown it implies that there exists a beta > 0 such that D(G)=O(\G\/(log \G\)(beta)) for all G. (C) 1995 Academic Press, Inc.