The {1, s}-weighted Davenport constant IN Ckn.

Abstract

Let G be a finite abelian group and let ∅ 6= A ⊂ Z. The A-weighted Davenport constant of G is the smallest positive integer DA(G) such that every sequence x1 · . . . · xDA(G) over G has a non-empty subsequence (xji )i such that ε1xj1 + ε2xj2 + . . . + εtxjt = 0 for some ε1, ε2, . . ., εt ∈ A. In this paper, we obtain both upper and lower bounds for D{1,s}(Ck n), where Cn denotes the cyclic group of order n, s2 ≡ 1 (mod n) and s 6≡ ±1 (mod n). These bounds become sharp in some “small” cases.