A variant of Davenport's constant

Abstract

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq D(G)/|A|$ be any integer where D(G) denotes the well-known Davenport's constant. In this article, we prove that for any sequence g_1, g_2, ..., g_k (not necessarily distinct) in G, one can always extract a subsequence g_{i_1}, g_{i_2}, ..., g_{i_\ell} with $1\leq \ell \leq k$ such that \begin{equation*} \sum_{j=1}^\ell a_{j}g_{i_j} = 0 {in} G, \end{equation*} where a_j \in A for all j. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem~1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.