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 $l \in N$ such that every sequence $S$ over $G$ of length $∣ S ∣ \geq l$ has a non-empty zero-sum subsequence. <a href="#GG2006-def-D">[GG2006-def-D]</a>

For $n \geq 2$ , let $C_{n}$ denote the cyclic group of order $n$ , and write

$C_{n}^{3} := C_{n} \oplus C_{n} \oplus C_{n} .$

We define

$C_{53} := n \geq 2 sup \frac{D ( C _{n}^{3} ) - 1}{n - 1},$

the maximal normalized Davenport constant among the rank- $3$ groups $C_{n}^{3}$ .

The following explicit uniform inequality is obtained from the inductive method and known multi-wise Davenport estimates over elementary prime groups:

$3 (n - 1) + 1 \leq D (C_{n}^{3}) \leq 4 n - P (n) - 2 (n \geq 2),$

where

$P (n) = p^{a} ∥ n max p^{a}$

is the largest primary component of $n$ . Since $P (n) \geq 2$ , this gives

$D (C_{n}^{3}) \leq 4 n - 4 = 4 (n - 1),$

and hence

$C_{53} \leq 4.$

The previously recorded published bounds included

$3 (n - 1) + 1 \leq D (C_{n}^{3}) \leq min {20369, 3^{ω (n)}} (n - 1) + 1,$

where $ω (n)$ is the number of distinct prime factors of $n$ . <a href="#Zak2019-omega-def">[Zak2019-omega-def]</a> <a href="#Zak2019-cor3.11">[Zak2019-cor3.11]</a> Here, the $3^{ω (n)}$ term is inherited from earlier published work, while the uniform numeric constant $20369$ is the explicit contribution in <a href="#Zak2019">[Zak2019]</a>. <a href="#Zak2019-prev-3omega">[Zak2019-prev-3omega]</a> <a href="#CMMPT2012">[CMMPT2012]</a>

In particular,

$3 \leq C_{53} \leq 4.$

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

$D (C_{n}^{3}) = 3 (n - 1) + 1,$

equivalently $C_{53} = 3$ . <a href="#GG2006-conj3.5">[GG2006-conj3.5]</a> <a href="#GG2006-D-equals-1-plus-d">[GG2006-D-equals-1-plus-d]</a> <a href="#GG2006-def-dstar">[GG2006-def-dstar]</a>

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 $D (C_{n}^{3}) = 3 (n - 1) + 1$ . <a href="#GG2006-thm3.1">[GG2006-thm3.1]</a> <a href="#GG2006-D-equals-1-plus-d">[GG2006-D-equals-1-plus-d]</a> <a href="#GG2006-def-dstar">[GG2006-def-dstar]</a>

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. <a href="#Zak2019-open-rank3">[Zak2019-open-rank3]</a>

Published work before the 2019 preprint includes Gao (2000) on rank- $3$ groups and Chintamani--Moriya--Gao--Paul--Thangadurai (2012), which gives the bound $D (C_{n}^{3}) \leq 3^{ω (n)} (n - 1) + 1$ used in Corollary 3.11. <a href="#Zak2019-ref-Gao2000">[Zak2019-ref-Gao2000]</a> <a href="#Zak2019-prev-3omega">[Zak2019-prev-3omega]</a> <a href="#CMMPT2012">[CMMPT2012]</a> <a href="#Gao2000">[Gao2000]</a>

Known upper bounds

Bound	Reference	Comments
$20369$	<a href="#Zak2019">[Zak2019]</a>	From Corollary 3.11: $D (C_{n}^{3}) \leq 20369 (n - 1) + 1$ for all $n \geq 2$ , hence $C_{53} \leq 20369$ . <a href="#Zak2019-cor3.11">[Zak2019-cor3.11]</a>
$4$	<a href="#Grinsztajn2026">[Grinsztajn2026]</a>	From the pointwise estimate $D (C_{n}^{3}) \leq 4 n - P (n) - 2$ , where $P (n) = max_{p^{a} ∥ n} p^{a}$ . Since $P (n) \geq 2$ , this gives $C_{53} \leq 4$ .

Known lower bounds

Bound	Reference	Comments
$3$	<a href="#GG2006">[GG2006]</a>	Using $d (C_{n}^{3}) \geq d^{\*} (C_{n}^{3}) = 3 (n - 1)$ and $D (G) = 1 + d (G)$ gives $D (C_{n}^{3}) \geq 3 (n - 1) + 1$ , hence $C_{53} \geq 3$ . <a href="#GG2006-d-ge-dstar">[GG2006-d-ge-dstar]</a> <a href="#GG2006-D-equals-1-plus-d">[GG2006-D-equals-1-plus-d]</a> <a href="#GG2006-def-dstar">[GG2006-def-dstar]</a>

Additional comments and links

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

Trivial lower bound. For $G ≅ C_{n_{1}} \oplus \dots \oplus C_{n_{r}}$ with $1 < n_{1} ∣ \dots ∣ n_{r}$ , 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 $D^{\*} (C_{n}^{3}) = 3 (n - 1) + 1$ . <a href="#GG2006-def-dstar">[GG2006-def-dstar]</a>

Wikipedia page on the Davenport constant

References

<a id="GG2006"></a>[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
<a id="GG2006-def-Cn"></a>[GG2006-def-Cn] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “For $n \in N$ , let $C_{n}$ denote a cyclic group with $n$ elements.”
<a id="GG2006-def-dstar"></a>[GG2006-def-dstar] loc: Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” quote: “ $d^{\*} (G) = \sum_{i = 1}^{r} (n_{i} - 1)$ .”
<a id="GG2006-def-D"></a>[GG2006-def-D] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “the smallest integer $l \in N$ such that every sequence $S$ over $G$ of length $∣ S ∣ \geq l$ has a non-empty zero-sum subsequence.”
<a id="GG2006-D-equals-1-plus-d"></a>[GG2006-D-equals-1-plus-d] loc: Expositiones Mathematicae PDF p.4, Definition 2.1 quote: “ $1 + d (G) = D (G)$ .”
<a id="GG2006-thm3.1"></a>[GG2006-thm3.1] loc: Expositiones Mathematicae PDF p.5, Theorem 3.1 quote: “If $G$ is a $p$ -group or $r (G) \leq 2$ , then $d (G) = d^{\*} (G)$ .”
<a id="GG2006-d-ge-dstar"></a>[GG2006-d-ge-dstar] loc: Expositiones Mathematicae PDF p.5, Section 3, just before Theorem 3.1 quote: “the crucial inequality $d (G) \geq d^{\*} (G)$ .”
<a id="GG2006-conj3.5"></a>[GG2006-conj3.5] loc: Expositiones Mathematicae PDF p.5, Section 3, Conjecture 3.5 quote: “If $G = C_{n}^{r}$ , where $n, r \in N_{\geq 3}$ , or $r (G) = 3$ , then $d (G) = d^{\*} (G)$ .”

<a id="Zak2019"></a>[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
<a id="Zak2019-open-rank3"></a>[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.”
<a id="Zak2019-omega-def"></a>[Zak2019-omega-def] loc: arXiv v1 PDF p.5, Corollary 3.11 quote: “let $ω (n)$ denote the number of distinct prime factors of $n$ .”
<a id="Zak2019-cor3.11"></a>[Zak2019-cor3.11] loc: arXiv v1 PDF p.5, Corollary 3.11 (Eq. (17)) quote: “ $3 (n - 1) + 1 \leq D (C_{n}^{3}) \leq min 20369, 3^{ω (n)} (n - 1) + 1$ .”
<a id="Zak2019-prev-3omega"></a>[Zak2019-prev-3omega] loc: arXiv v1 PDF p.5, proof of Corollary 3.11 quote: “By [3, Theorem 1.2], we get $D (C_{n}^{3}) \leq 3^{ω (n)} (n - 1) + 1$ .”
<a id="Zak2019-ref-Gao2000"></a>[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.”

<a id="CMMPT2012"></a>[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

<a id="Gao2000"></a>[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

<a id="Grinsztajn2026"></a>[Grinsztajn2026] Grinsztajn, Max. An upper bound for the Davenport constant of $C_{n}^{3}$ . Proof PDF: https://github.com/maaxgrin/davenport-cn3-bound/blob/main/davenport_cn3_bound.pdf

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

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Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem