Davenport constant with weights

TL;DR

The paper proves that for the cyclic group \(\mathbb{Z}/n\mathbb{Z}\) and any non-empty weight set \(A\subseteq \mathbb{Z}\), the weighted EGZ constant satisfies \(E_A(n)=D_A(n)+n-1\). This resolves a conjecture raised by Adhikari–Rath, Adhikari–Chen, Thangadurai, and Griffiths.

Key findings

• The main theorem is \(E_A(n)=D_A(n)+n-1\) for any non-empty set \(A\subseteq \mathbb{Z}\) (Theorem 1.2), proving Conjecture 1.1.
• The result settles the problem raised by Adhikari and Rath, Adhikari and Chen, Thangadurai, and Griffiths.
• The proof uses the main theorem of Devos, Goddyn, and Mohar (Theorem DGM) and a recently proved theorem of the authors (Theorem YZ) on Davenport’s constant.
• The paper notes that equality was previously proved in special cases: \(A=\{1\}\), \(\{-1,1\}\), \(\mathbb{Z}_n^\star\), \(A=(a_1,\dots,a_r)\) with \(\gcd(a_2-a_1,\dots,a_r-a_1,n)=1\), or \(n=p\) prime.

Methods

• Theoretical proof in additive combinatorics/zero-sum theory for the cyclic group \(G=\mathbb{Z}/n\mathbb{Z}\); it defines \(D_A(n)\) and \(E_A(n)\) for sequences in \(G\) with weights from \(A\).
• The proof shows \(E_A(n)\ge D_A(n)+n-1\) is easy, then proves the reverse inequality by contradiction using a minimal counterexample \((A,G,S)\), where \(t=D_A(n)+n-1\) and \(S=x_1\cdots x_t\), aiming to show \(0\in\sum_n(A)\).
• The argument uses stabilization \(H=\operatorname{stab}(\sum_n(A))\), Theorem DGM, Theorem YZ, a claim \(D_A(G)\ge D_A(H_1)\ge D(H)=|H|\), and a case split on \(s\ge n\) versus \(s<n\).

Limitations

• The paper states no limitations.
• Evident from the text: the theorem is proved for cyclic groups \(G=\mathbb{Z}/n\mathbb{Z}\); the abstract and proof are framed for cyclic \(G\), not general finite abelian groups.
• Evident from the text: the proof depends on Theorem YZ, cited as “submitted” ([13]), and on Theorem DGM (Devos–Goddyn–Mohar, “to appear” in the references).