The Weighted Davenport constant of a group and a related extremal problem II

TL;DR

This paper investigates the extremal function related to the weighted Davenport constant in finite abelian groups, providing new bounds and confirming conjectures for prime groups, especially for large primes.

Contribution

It establishes an improved upper bound for the extremal function (p,k) in prime groups, advancing understanding of weighted Davenport constants.

Findings

Proves (p,k)  4^{k^2} p^{1/k} for large primes p.

Confirms the conjecture (p,k) =   p^{1/k} for sufficiently large primes.

Provides bounds that bridge previous known inequalities for the extremal function.

Abstract

For a finite abelian group $G$ with $\exp(G)=n$ and an integer $k\ge 2$, Balachandran and Mazumdar \cite{BM} introduced the extremal function $\fD_G(k)$ which is defined to be $\min\{|A|: \emptyset \neq A\subseteq[1,n-1]\textrm{\ with\ }D_A(G)\le k\}$ (and $\infty$ if there is no such $A$), where $D_A(G)$ denotes the $A$-weighted Davenport constant of the group $G$. Denoting $\fD_G(k)$ by $\fD(p,k)$ when $G=\bF_p$ (for $p$ prime), it is known (\cite{BM}) that $p^{1/k}-1\le \fD(p,k)\le O_k(p\log p)^{1/k}$ holds for each $k\ge 2$ and $p$ sufficiently large, and that for $k=2,4$, we have the sharper bound $\fD(p,k)\le O(p^{1/k})$. It was furthermore conjectured that $\fD(p,k)=\Theta(p^{1/k})$. In this short paper we prove that $\fD(p,k)\le 4^{k^2}p^{1/k}$ for sufficiently large primes $p$.