Davenport constant with weights

For the cyclic group G = Z/nZ and any non-empty A ∈ Z. We define the Davenport constant of G with weight A, denoted by DA(n), to be the least natural number k such that for any sequence (x1, · · · , xk) with xi ∈ G, there exists a non-empty subsequence (xj1, · · · , xjl) and a1, · · · , al ∈ A such that ∑l i=1 aixji = 0. Similarly, we define the constant EA(n) to be the least t ∈ N such that for all sequences (x1, · · · , xt) with xi ∈ G, there exist indices j1, · · · , jn ∈ N, 1 ≤ j1 < · · · < jn ≤ t, and θ1, · · · , θn ∈ A with ∑n i=1 θixji = 0. In the present paper, we show that EA(n) = DA(n)+n−1. This solve the problem raised by Adhikari and Rath [3], Adhikari and Chen [2], Thangadurai [12] and Griffiths [10]. MSC: 11B50