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

Let G≃Z/k1Z⊕⋯⊕Z/kNZG≃Z/k1Z⊕⋯⊕Z/kNZ be a finite abelian group with ki|ki−1(2≤i≤N). For a matrix Y=(ai,j)∈ZR×SY=(ai,j)∈ZR×S satisfying ai,1+⋯+ai,S=0(1≤i≤R), let DY(G)DY(G) denote the maximal cardinality of a set A⊆GA⊆G for which the equations ai,1x1+⋯+ai,SxS=0(1≤i≤R) are never satisfied simultaneously by distinct elements x1,…,xS∈Ax1,…,xS∈A. Under certain assumptions on YY and GG, we prove an upper bound of the form DY(G)≤|G|(C/N)γDY(G)≤|G|(C/N)γ for positive constants CC and γγ.