An upper bound for the kk-barycentric Davenport constant of groups of prime order

Let GG be a finite abelian group and let k⩾2k⩾2 be an integer. A sequence of kk elements a1,a2,…,aka1,a2,…,ak in GG is called a kk-barycentric sequence if there exists j∈{1,2,…,k}j∈{1,2,…,k} such that ∑i=1kai=kaj. The kk-barycentric Davenport constant BD(k,G) is defined to be the smallest number ss such that every sequence in GG of length ss contains a kk-barycentric subsequence. In this paper, we prove that if p⩾5p⩾5 is a prime, then BD(k,Zp)⩽p+k−⌊p−2k⌋−2 for 3⩽k⩽p−13⩽k⩽p−1, which improves a result of Delorme et al.