A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression (II)

Let G similar or equal to Z/k(1)Z circle plus ... circle plus Z/k(N)Z be a finite abelian group with k(i)vertical bar k(i-1) (2 <= i <= N). For a matrix Y = (a(i,j)) is an element of Z(RxS) satisfying a(i,1) + ... + a(i,S) = 0 (1 <= i <= R), let D(Y)(G) denote the maximal cardinality of a set A subset of G for which the equations a(i,1)x(1) + ... + a(i,S)x(S) = 0 (1 <= i <= R) are never satisfied simultaneously by distinct elements x(1), ..., x(S) is an element of A. Under certain assumptions on Y and G, we prove an upper bound of the form D(Y)(G) <= |G|(C/N)(gamma) for positive constants C and gamma. (C) 2010 Elsevier Ltd. All rights reserved.