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.