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

Let G ' Z/k1Z⊕ · · · ⊕ Z/kNZ be a finite abelian group with ki|ki−1 (2 ≤ i ≤ N). For a matrix Y = ( ai,j ) ∈ ZR×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) for positive constants C and γ.