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

Abstract

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+· · · +rs xs = 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.