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

Let G ≃ Z / k 1 Z ⊕ ⋯ ⊕ Z / k N Z be a finite abelian group with k i | k i − 1 ( 2 ≤ i ≤ N ) . For a matrix Y = ( a i , j ) ∈ Z R × S satisfying a i , 1 + ⋯ + a i , S = 0 ( 1 ≤ i ≤ R ) , let D Y ( G ) denote the maximal cardinality of a set A ⊆ G for which the equations a i , 1 x 1 + ⋯ + a i , S x S = 0 ( 1 ≤ i ≤ R ) are never satisfied simultaneously by distinct elements x 1 , … , x S ∈ A . Under certain assumptions on Y and G , we prove an upper bound of the form D Y ( G ) ≤ | G | ( C / N ) γ for positive constants C and γ .