Davenport constant for $C_n^3$

Description of constant

In zero-sum theory, the Davenport constant $D(G)$ of a finite abelian group $G$ is defined as the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $\lvert S\rvert\ge l$ has a non-empty zero-sum subsequence. [GG2006-def-D]

For $n\ge 2$, let $C_n$ denote the cyclic group of order $n$, and write

[GG2006-def-Cn]

We define

the maximal normalized Davenport constant among the rank-$3$ groups $C_n^3$.

The best general bounds currently available in this setting include the explicit uniform inequality

where $\omega(n)$ is the number of distinct prime factors of $n$. [Zak2019-omega-def] [Zak2019-cor3.11]

Here, the $3^{\omega(n)}$ term is inherited from earlier published work, while the uniform numeric constant $20369$ is the new explicit contribution in [Zak2019]. [Zak2019-prev-3omega] [CMMPT2012]

In particular,

[Zak2019-cor3.11]

A long-standing conjecture (in a stronger, pointwise form) predicts that for all $n\ge 2$ one has

equivalently $C_{53a}=3$. [GG2006-conj3.5] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]

One unconditional family of exact evaluations is given by prime powers: if $n$ is a prime power, then $C_n^3$ is a $p$-group, and Theorem 3.1 implies $d(C_n^3)=d^*(C_n^3)=3(n-1)$ and hence $D(C_n^3)=3(n-1)+1$. [GG2006-thm3.1] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]

The general determination of $D(G)$ for rank-$3$ groups (and in particular the pointwise determination of $D(C_n^3)$ for all $n$) remains open. [Zak2019-open-rank3]

Published work before the 2019 preprint includes Gao (2000) on rank-$3$ groups and Chintamani–Moriya–Gao–Paul–Thangadurai (2012), which gives the bound $D(C_n^3)\le 3^{\omega(n)}(n-1)+1$ used in Corollary 3.11. [Zak2019-ref-Gao2000] [Zak2019-prev-3omega] [CMMPT2012] [Gao2000]

Known upper bounds

Known lower bounds

Additional comments and links

• Zero-sumfree reformulation. If $d(G)$ denotes the maximal length of a zero-sumfree sequence over $G$, then $D(G)=1+d(G) (Definition 2.1 in [GG2006]). In these terms, the conjecture for $C_n^3$ is $d(C_n^3)=3(n-1)$.
  [GG2006-D-equals-1-plus-d] [GG2006-conj3.5]

• Trivial lower bound. For $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$, the quantity $d^(G)=\sum_{i=1}^r(n_i-1)$ is the standard lower bound for $d(G)$; for $C_n^3$ this gives $d^(C_n^3)=3(n-1)$ and $D^*(C_n^3)=3(n-1)+1$.
  [GG2006-def-dstar]

• Wikipedia page on the Davenport constant

References

• [GG2006] Gao, Weidong; Geroldinger, Alfred. Zero-sum problems in finite abelian groups: A survey. Expositiones Mathematicae 24 (2006), 337–369. DOI: https://doi.org/10.1016/j.exmath.2006.07.002. Publisher entry (DOI). Mirror PDF. Google Scholar
  • [GG2006-def-Cn] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “For $n\in\mathbb{N}$, let $C_n$ denote a cyclic group with $n$ elements.”
  • [GG2006-def-dstar] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “$d^*(G)=\sum_{i=1}^r (n_i-1)$.”
  • [GG2006-def-D] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “the smallest integer $l\in\mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\ge l$ has a non-empty zero-sum subsequence.”
  • [GG2006-D-equals-1-plus-d] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “$1+d(G)=D(G)$.”
  • [GG2006-thm3.1] loc: Expositiones Mathematicae PDF p.5, Theorem 3.1 quote: “If $G$ is a $p$-group or $r(G)\le 2$, then $d(G)=d^*(G)$.”
  • [GG2006-d-ge-dstar] loc: Expositiones Mathematicae PDF p.5, Section 3, just before Theorem 3.1 quote: “the crucial inequality $d(G)\ge d^*(G)$.”
  • [GG2006-conj3.5] loc: Expositiones Mathematicae PDF p.5, Section 3, Conjecture 3.5 quote: “If $G=C_n^r$, where $n,r\in\mathbb{N}_{\ge 3}$, or $r(G)=3$, then $d(G)=d^*(G)$.”
• [Zak2019] Zakarczemny, Maciej. Note on the Davenport’s constant for finite abelian groups with rank three. (2019). PDF: https://arxiv.org/pdf/1910.10984. DOI: https://doi.org/10.48550/arXiv.1910.10984. Google Scholar
  • [Zak2019-open-rank3] loc: arXiv v1 PDF p.1, Introduction quote: “The exact value of the Davenport constant for groups of rank three is still unknown and this is an open and well-studied problem.”
  • [Zak2019-omega-def] loc: arXiv v1 PDF p.5, Corollary 3.11 quote: “let $\omega(n)$ denote the number of distinct prime factors of $n$.”
  • [Zak2019-cor3.11] loc: arXiv v1 PDF p.5, Corollary 3.11 (Eq. (17)) quote: “$3(n-1)+1\le D(C_n^3)\le \min{20369,3^{\omega(n)}}(n-1)+1$.”
  • [Zak2019-prev-3omega] loc: arXiv v1 PDF p.5, proof of Corollary 3.11 quote: “By [3, Theorem 1.2], we get $D(C_n^3)\le 3^{\omega(n)}(n-1)+1$.”
  • [Zak2019-ref-Gao2000] loc: arXiv v1 PDF p.7, References [9] quote: “[9] W. D. Gao, On Davenport’s constant of finite abelian groups with rank three, Discrete Mathematics 222 (2000), pages 111-124.”

• [CMMPT2012] Chintamani, M. N.; Moriya, B. K.; Gao, W. D.; Paul, P.; Thangadurai, R. New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants. Archiv der Mathematik 98 (2012), no. 2, 133–142. DOI: https://doi.org/10.1007/s00013-011-0345-z. Google Scholar

• [Gao2000] Gao, W. D. On Davenport’s constant of finite abelian groups with rank three. Discrete Mathematics 222 (2000), no. 1–3, 111–124. DOI: https://doi.org/10.1016/S0012-365X(00)00010-8. Google Scholar

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.