The Weighted Davenport Constant of a group and a related extremal problem

TL;DR

This paper investigates the weighted Davenport Constant in finite abelian groups, introduces a new extremal problem related to the minimal size of subsets ensuring bounded weighted zero-sum sequences, and provides near-optimal bounds for cyclic groups of prime order.

Contribution

It formulates and analyzes a novel extremal problem for the weighted Davenport Constant, especially for cyclic groups of prime order, with asymptotic bounds for specific values of k.

Findings

Established near-optimal bounds for all large primes p.

Derived asymptotically tight bounds for k=2 and k=4.

Introduced a new extremal problem related to weighted zero-sum sequences.

Abstract

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary $G$-sequence $(x_1,\ldots,x_k)$, there exists a non-empty subsequence $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$: For an integer $k\ge 2$, determine $\fD_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ `not-too-small', this is a well-posed problem and one of the most interesting cases occurs for $G=\Z_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.