A variant of Davenport's constant

TL;DR

This paper extends Davenport's constant for finite abelian p-groups by establishing conditions under which specific weighted zero-sum subsequences exist, using algebraic and number theory methods.

Contribution

It introduces a new variant of Davenport's constant involving weighted sums with elements from a subset A, providing bounds and sharp results for cyclic groups.

Findings

Established a generalized zero-sum subsequence existence condition.

Provided examples showing the bounds are optimal.

Proved sharp results for cyclic groups.

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.