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

This is a post-peer-review, pre-copyedit version of an article published in Designs, Codes and Cryptography. The final authenticated version is available online at: https://doi.org/10.1007/s10623-009-9268-0Let 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

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

This article is made available through Elsevier's Open Archive, https://doi.org/10.1016/j.ejc.2010.09.008. © 2010 Elsevier Ltd. All rights reserved.Let G ≃ Z/k1Z ⊕ · · · ⊕ Z/kN Z be a finite abelian group with ki |ki−1 (2 ≤ i ≤ N). For a matrix Y = (ai,j) ∈ Z R×S satisfying ai,1 + · · · + ai,S = 0 (1 ≤ i ≤ R), let DY (G) denote the maximal cardinality of a set A ⊆ G for which the equations ai,1x1 + · · · + ai,SxS = 0 (1 ≤ i ≤ R) are never satisfied simultaneously by distinct elements x1, . . . , xS ∈ A. Under certain assumptions on Y and G, we prove an upper bound of the form DY (G) ≤ |G|(C/N) γ for positive constants C and γ