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

Abstract

Let G ≃ Z/k1Z ⊕ · · · ⊕ Z/kN Z be a finite abelian group with ki |ki−1 (2 ≤ i ≤ N). For a matrix Y = (ai,j) ∈ Z R×S satisfying ai,1 + · · · + ai,S = 0 (1 ≤ i ≤ R), let DY (G) denote the maximal cardinality of a set A ⊆ G for which the equations ai,1x1 + · · · + ai,SxS = 0 (1 ≤ i ≤ R) are never satisfied simultaneously by distinct elements x1, . . . , xS ∈ A. Under certain assumptions on Y and G, we prove an upper bound of the form DY (G) ≤ |G|(C/N) γ for positive constants C and γ .