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

Abstract

Let r 1, …, r s be non-zero integers satisfying r 1 + ⋯ + r s = 0. Let G $${\simeq \mathbb{Z} / k_1 \mathbb{Z}\oplus \cdots \oplus \mathbb{Z} / k_n \mathbb{Z}}$$ be a finite abelian group with k i |k i-1(2 ≤ i ≤ n), and suppose that (r i , k 1) = 1(1 ≤ i ≤ s). Let $${D_{\mathbf r}(G)}$$ denote the maximal cardinality of a set $${A \subseteq G}$$ which contains no non-trivial solution of r 1 x 1 + ⋯ + r s x s = 0 with $${x_i\,\in\,A (1 \le i \le s)}$$ . We prove that $${D_{\mathbf r}(G) \ll |G|/n^{s-2}}$$ . We also apply this result to study problems in finite projective spaces.