Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A ⊆ Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by D A ( G ) , to be the least natural number k such that for any sequence ( x 1 , … , x k ) with x i ∈ G , there exists a non-empty subsequence ( x j 1 , … , x j l ) and a 1 , … , a l ∈ A such that ∑ i = 1 l a i x j i = 0 . Similarly, for any such set A, E A ( G ) is defined to be the least t ∈ N such that for all sequences ( x 1 , … , x t ) with x i ∈ G , there exist indices j 1 , … , j n ∈ N , 1 ⩽ j 1 < ⋯ < j n ⩽ t , and ϑ 1 , … , ϑ n ∈ A with ∑ i = 1 n ϑ i x j i = 0 . In the present paper, we establish a relation between the constants D A ( G ) and E A ( G ) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G = Z / n Z and the relation we establish had been conjectured in that particular case.