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

Let r1, . . . , rs be non-zero integers satisfying r1 + · · ·+ rs = 0. Let G ' Z/k1Z⊕ · · · ⊕ Z/knZ be a finite abelian group with ki|ki−1 (2 ≤ i ≤ n), and suppose that (ri, k1) = 1 (1 ≤ i ≤ s). Let Dr(G) denote the maximal cardinality of a set A ⊆ G which contains no non-trivial solution of r1x1 + · · · + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that Dr(G) |G|/ns−2. We also apply this result to study problems in finite projective spaces.