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

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