|
| 1 | +# Davenport constant for $C_n^3$ |
| 2 | + |
| 3 | +## Description of constant |
| 4 | + |
| 5 | +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. |
| 6 | +<a href="#GG2006-def-D">[GG2006-def-D]</a> |
| 7 | + |
| 8 | +For $n\ge 2$, let $C_n$ denote the cyclic group of order $n$, and write |
| 9 | + |
| 10 | +$$ |
| 11 | +C_n^3\ :=\ C_n\oplus C_n\oplus C_n. |
| 12 | +$$ |
| 13 | + |
| 14 | +<a href="#GG2006-def-Cn">[GG2006-def-Cn]</a> |
| 15 | + |
| 16 | +We define |
| 17 | + |
| 18 | +$$ |
| 19 | +C_{53a}\ :=\ \sup_{n\ge 2}\ \frac{D(C_n^3)-1}{n-1}, |
| 20 | +$$ |
| 21 | + |
| 22 | +the maximal normalized Davenport constant among the rank-$3$ groups $C_n^3$. |
| 23 | + |
| 24 | +The best general bounds currently available in this setting include the explicit uniform inequality |
| 25 | + |
| 26 | +$$ |
| 27 | +3(n-1)+1\ \le\ D(C_n^3)\ \le\ \min\{20369,\ 3^{\omega(n)}\}(n-1)+1 |
| 28 | +\qquad (n\ge 2), |
| 29 | +$$ |
| 30 | + |
| 31 | +where $\omega(n)$ is the number of distinct prime factors of $n$. |
| 32 | +<a href="#Zak2019-omega-def">[Zak2019-omega-def]</a> <a href="#Zak2019-cor3.11">[Zak2019-cor3.11]</a> |
| 33 | + |
| 34 | +Here, the $3^{\omega(n)}$ term is inherited from earlier published work, while the uniform numeric constant $20369$ is the new explicit contribution in <a href="#Zak2019">[Zak2019]</a>. |
| 35 | +<a href="#Zak2019-prev-3omega">[Zak2019-prev-3omega]</a> <a href="#CMMPT2012">[CMMPT2012]</a> |
| 36 | + |
| 37 | +In particular, |
| 38 | + |
| 39 | +$$ |
| 40 | +3\ \le\ C_{53a}\ \le\ 20369. |
| 41 | +$$ |
| 42 | + |
| 43 | +<a href="#Zak2019-cor3.11">[Zak2019-cor3.11]</a> |
| 44 | + |
| 45 | +A long-standing conjecture (in a stronger, pointwise form) predicts that for all $n\ge 2$ one has |
| 46 | + |
| 47 | +$$ |
| 48 | +D(C_n^3)\ =\ 3(n-1)+1, |
| 49 | +$$ |
| 50 | + |
| 51 | +equivalently $C_{53a}=3$. |
| 52 | +<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> |
| 53 | + |
| 54 | +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 $D(C_n^3)=3(n-1)+1$. |
| 55 | +<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> |
| 56 | + |
| 57 | +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. |
| 58 | +<a href="#Zak2019-open-rank3">[Zak2019-open-rank3]</a> |
| 59 | + |
| 60 | +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. |
| 61 | +<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> |
| 62 | + |
| 63 | +## Known upper bounds |
| 64 | + |
| 65 | +| Bound | Reference | Comments | |
| 66 | +| ----- | --------- | -------- | |
| 67 | +| $20369$ | <a href="#Zak2019">[Zak2019]</a> | From Corollary 3.11: $D(C_n^3)\le 20369(n-1)+1$ for all $n\ge 2$, hence $C_{53a}\le 20369$. <a href="#Zak2019-cor3.11">[Zak2019-cor3.11]</a> | |
| 68 | + |
| 69 | +## Known lower bounds |
| 70 | + |
| 71 | +| Bound | Reference | Comments | |
| 72 | +| ----- | --------- | -------- | |
| 73 | +| $3$ | <a href="#GG2006">[GG2006]</a> | Using $d(C_n^3)\ge d^*(C_n^3)=3(n-1)$ and $D(G)=1+d(G)$ gives $D(C_n^3)\ge 3(n-1)+1$, hence $C_{53a}\ge 3$. <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> | |
| 74 | + |
| 75 | +## Additional comments and links |
| 76 | + |
| 77 | +- **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 $C_n^3$ is $d(C_n^3)=3(n-1)$. |
| 78 | + <a href="#GG2006-D-equals-1-plus-d">[GG2006-D-equals-1-plus-d]</a> <a href="#GG2006-conj3.5">[GG2006-conj3.5]</a> |
| 79 | + |
| 80 | +- **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 $D^*(C_n^3)=3(n-1)+1$. |
| 81 | + <a href="#GG2006-def-dstar">[GG2006-def-dstar]</a> |
| 82 | + |
| 83 | +- [Wikipedia page on the Davenport constant](https://en.wikipedia.org/wiki/Davenport_constant) |
| 84 | + |
| 85 | +## References |
| 86 | + |
| 87 | +- <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)](https://doi.org/10.1016/j.exmath.2006.07.002). [Mirror PDF](https://cfc.nankai.edu.cn/_upload/article/files/c6/e1/a2c52bf04b1896f59003b5993582/5c9e49ea-af5b-44ac-b153-5dbb6d8ae9a3.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Gao+Geroldinger+Zero-sum+problems+in+finite+abelian+groups%3A+a+survey+Expositiones+Mathematicae+24+2006+337%E2%80%93369) |
| 88 | + - <a id="GG2006-def-Cn"></a>**[GG2006-def-Cn]** |
| 89 | + **loc:** Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” |
| 90 | + **quote:** “For $n\in\mathbb{N}$, let $C_n$ denote a cyclic group with $n$ elements.” |
| 91 | + - <a id="GG2006-def-dstar"></a>**[GG2006-def-dstar]** |
| 92 | + **loc:** Expositiones Mathematicae PDF p.2, Section 2 “Preliminaries” |
| 93 | + **quote:** “$d^*(G)=\sum_{i=1}^r (n_i-1)$.” |
| 94 | + - <a id="GG2006-def-D"></a>**[GG2006-def-D]** |
| 95 | + **loc:** Expositiones Mathematicae PDF p.4, Definition 2.1 |
| 96 | + **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.” |
| 97 | + - <a id="GG2006-D-equals-1-plus-d"></a>**[GG2006-D-equals-1-plus-d]** |
| 98 | + **loc:** Expositiones Mathematicae PDF p.4, Definition 2.1 |
| 99 | + **quote:** “$1+d(G)=D(G)$.” |
| 100 | + - <a id="GG2006-thm3.1"></a>**[GG2006-thm3.1]** |
| 101 | + **loc:** Expositiones Mathematicae PDF p.5, Theorem 3.1 |
| 102 | + **quote:** “If $G$ is a $p$-group or $r(G)\le 2$, then $d(G)=d^*(G)$.” |
| 103 | + - <a id="GG2006-d-ge-dstar"></a>**[GG2006-d-ge-dstar]** |
| 104 | + **loc:** Expositiones Mathematicae PDF p.5, Section 3, just before Theorem 3.1 |
| 105 | + **quote:** “the crucial inequality $d(G)\ge d^*(G)$.” |
| 106 | + - <a id="GG2006-conj3.5"></a>**[GG2006-conj3.5]** |
| 107 | + **loc:** Expositiones Mathematicae PDF p.5, Section 3, Conjecture 3.5 |
| 108 | + **quote:** “If $G=C_n^r$, where $n,r\in\mathbb{N}_{\ge 3}$, or $r(G)=3$, then $d(G)=d^*(G)$.” |
| 109 | + |
| 110 | +- <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](https://scholar.google.com/scholar?q=Zakarczemny+Note+on+the+Davenport%E2%80%99s+constant+for+finite+abelian+groups+with+rank+three) |
| 111 | + - <a id="Zak2019-open-rank3"></a>**[Zak2019-open-rank3]** |
| 112 | + **loc:** arXiv v1 PDF p.1, Introduction |
| 113 | + **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.” |
| 114 | + - <a id="Zak2019-omega-def"></a>**[Zak2019-omega-def]** |
| 115 | + **loc:** arXiv v1 PDF p.5, Corollary 3.11 |
| 116 | + **quote:** “let $\omega(n)$ denote the number of distinct prime factors of $n$.” |
| 117 | + - <a id="Zak2019-cor3.11"></a>**[Zak2019-cor3.11]** |
| 118 | + **loc:** arXiv v1 PDF p.5, Corollary 3.11 (Eq. (17)) |
| 119 | + **quote:** “$3(n-1)+1\le D(C_n^3)\le \min\{20369,3^{\omega(n)}\}(n-1)+1$.” |
| 120 | + - <a id="Zak2019-prev-3omega"></a>**[Zak2019-prev-3omega]** |
| 121 | + **loc:** arXiv v1 PDF p.5, proof of Corollary 3.11 |
| 122 | + **quote:** “By [3, Theorem 1.2], we get $D(C_n^3)\le 3^{\omega(n)}(n-1)+1$.” |
| 123 | + - <a id="Zak2019-ref-Gao2000"></a>**[Zak2019-ref-Gao2000]** |
| 124 | + **loc:** arXiv v1 PDF p.7, References [9] |
| 125 | + **quote:** “[9] W. D. Gao, On Davenport's constant of finite abelian groups with rank three, Discrete Mathematics 222 (2000), pages 111-124.” |
| 126 | + |
| 127 | +- <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](https://scholar.google.com/scholar?q=New+upper+bounds+for+the+Davenport+and+for+the+Erd%C5%91s-Ginzburg-Ziv+constants) |
| 128 | + |
| 129 | +- <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](https://scholar.google.com/scholar?q=On+Davenport%27s+constant+of+finite+abelian+groups+with+rank+three) |
| 130 | + |
| 131 | +## Contribution notes |
| 132 | + |
| 133 | +Prepared with assistance from ChatGPT 5.2 Pro. |
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