Zero-sum problems with subgroup weights
Given positive integers \(n\) and \(d\), let \(G=({\mathbb Z}_n)^d\) and \(A\subset{\mathbb Z}_n\setminus \{0\}\). Let \(D_A(n,d)\) be the least positive integer \(k\) such that for any sequence \(\{{\mathbf x}_1,\dots,{\mathbf x}_k\}\) of \(k\) elements in \(G\) there exists a non-empty subsequence \(\{{\mathbf x}_{j_1},\dots,{\mathbf x}_{j_t}\}\) and \(a_1,\dots,a_t\) in \(A\) such that \(\sum_{i=1}^t a_i{\mathbf x}_{j_i}={\mathbf 0}\). Similarly, \(f_A(n,d)\) is defined as the least positive integer \(k\) such that for any sequence \(\{{\mathbf x}_1,\dots,{\mathbf x}_k\}\) of \(k\) elements in \(G\) there exists a subsequence \(\{{\mathbf x}_{j_1},\dots,{\mathbf x}_{j_n}\}\) of length \(n\) and \(a_1,\dots,a_n\) in \(A\) with \(\sum_{i=1}^n a_i{\mathbf x}_{j_i}={\mathbf 0}\). These constants generalize the known Davenport constant \(D_A(n,1)\). The aim of the paper is to generalize previous results for \(D_A(n,1)\) and \(f_A(n,1)\) proved by the first author with other coauthors in [Proc. Indian Acad. Sci., Math. Sci. 118, No. 2, 183--188 (2008; Zbl 1207.11030), Integers 6, Paper A30, 6 p., electronic only (2006; Zbl 1107.11018), and Integers 8, No. 1, Article A52, 11 p., electronic only (2008; Zbl 1202.11018)] to general \(d\) with \(n=p\), a prime, and \(A\) a subgroup of \({\mathbb Z}_p^*\).
scientific article; zbMATH DE number 5810046
scientific article