Davenport's constant for finite abelian groups of rank three

For a finite abelian group $G$, Davenport's constant $D(G)$ is the least $n$ such that every sequence of $n$ elements of $G$ contains a nonempty zero-sum subsequence. In invariant-factor form $G\cong C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$ with $n_1\mid n_2\mid n_3$, the elementary construction of zero-sumfree sequences gives $D(G)\ge D^*(G)=n_1+n_2+n_3-2$. Equality is known for $p$-groups and for groups of rank at most two, and it fails for infinitely many groups of every rank at least four, leaving rank three the only undecided rank. The source paper (Bhowmik and Schlage-Puchta, 2006) proved $D(G)=D^*(G)$ for the family $C_3\oplus C_3\oplus C_{3d}$ and posed the general rank-three question, which they were inclined to believe true. Several further rank-three families have since been settled, but the general question — does $D(G)=D^*(G)$ hold for every rank-three finite abelian group, or does a counterexample exist? — remains open, with explicit statements of its openness in papers from 2024.

Background

By the structure theorem, every finite abelian group is a direct sum of cyclic groups $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $C_n=\mathbb{Z}/n\mathbb{Z}$ and $n_1\mid\cdots\mid n_r$; the number $r$ of cyclic summands is the rank of $G$. Additive combinatorics studies such groups through sequences (multisets) $S=g_1,\ldots,g_\ell$ of elements; a subsequence of $S$ is zero-sum if the sum of its elements is the identity. Davenport's constant $D(G)$ is the smallest $n$ such that every sequence of $n$ elements of $G$ contains a nonempty zero-sum subsequence — equivalently, one plus the maximal length of a zero-sumfree sequence, that is, a sequence containing no nonempty zero-sum subsequence. Taking a generator of each cyclic summand $n_i-1$ times gives a zero-sumfree sequence of length $\sum_i(n_i-1)$, hence the universal lower bound, written $D^*(G)$ (denoted $M(G)$ in the source paper):

$$D(G)\ \ge\ D^*(G)=\sum_{i=1}^{r}(n_i-1)+1.$$

Determining $D(G)$ exactly matters for the arithmetic of nonunique factorizations: for a Krull monoid — a cancellative monoid admitting a divisor theory, generalizing the multiplicative monoid of a ring of algebraic integers — with class group $G$, the possible factorization lengths are governed by the block monoid (the monoid of zero-sum sequences over $G$), whose structure is controlled by $D(G)$. It also feeds the Erdős–Ginzburg–Ziv constant of $G$ — the least length forcing a zero-sum subsequence of length exactly $\exp(G)$, the exponent (least common multiple of the element orders) of $G$ — and related zero-sum invariants; the standard reference is the survey of Gao and Geroldinger (2006).

The equality $D(G)=D^*(G)$ was conjectured in the 1960s for all finite abelian groups. Olson (1969) proved it for all finite $p$-groups (groups whose order is a power of a prime) and for all groups of rank at most two. Geroldinger and Schneider (1992) constructed infinitely many counterexamples of ranks four and five, and by the work of Edel, Elsholtz, Geroldinger, Kubertin and Rackham (2007) counterexamples exist in every rank $r\ge 4$; the source paper already records that infinitely many are known for every rank at least four. Rank three is thus the one remaining rank, and the question whether $D(G)=D^*(G)$ for every rank-three group goes back at least to the 1969 report of van Emde Boas and Kruyswijk, which flagged the case $C_3\oplus C_3\oplus C_{15}$.

Bhowmik and Schlage-Puchta (2006) settled that case by proving $D(G)=3d+4=D^*(G)$ for every group $C_3\oplus C_3\oplus C_{3d}$ (Delorme, Ordaz and Quiroz had previously obtained $D(G)\le D^*(G)+2$ for this family). Their method projects a sequence onto the small group $C_3^3$ and runs an exhaustive case analysis of when the projected multiset can carry labels in $C_d$ without forcing a zero-sum; the essential new cases are those with $(6,d)=1$, the others being previously known. They close by noting that the method applies in principle to further families $C_{a_1}\oplus\cdots\oplus C_{a_{r-1}}\oplus C_{a_r d}$ with $(d,a_r)=1$, but that the combinatorial effort grows rapidly, and they propose automation as future work. This automation strand was substantially realized by Bhowmik, Halupczok and Schlage-Puchta (2007) as a general inductive method with a decidability theorem, though one that does not settle the rank-three question. Girard and Schmid (2019) settled the complementary large family $C_2\oplus C_{2m}\oplus C_{2mn}$. Despite all this, for a general rank-three group $C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$ the exact value of $D(G)$ — in particular whether it can exceed $D^*(G)$ — remains unknown.

Problem Statement

Let $G$ be a finite abelian group of rank three, written in invariant-factor form

$$G\cong C_{n_1}\oplus C_{n_2}\oplus C_{n_3},\qquad 1<n_1\mid n_2\mid n_3,$$

where $C_n=\mathbb{Z}/n\mathbb{Z}$. Is it true that

$$D(G)=D^*(G)=n_1+n_2+n_3-2$$

for every such group, where $D(G)$ is Davenport's constant, the least $n$ such that every sequence of $n$ elements of $G$ contains a nonempty zero-sum subsequence? A failure of the equality would be witnessed by a rank-three finite abelian group $G$ together with a zero-sumfree sequence (a sequence containing no nonempty zero-sum subsequence) over $G$ of length $n_1+n_2+n_3-2$.

References

1. Gautami Bhowmik and Jan-Christoph Schlage-Puchta, Davenport's constant for groups of the form Z_3 ⊕ Z_3 ⊕ Z_{3d}, arXiv:math/0610416 (2006). https://arxiv.org/abs/math/0610416
2. Gautami Bhowmik, Immanuel Halupczok, and Jan-Christoph Schlage-Puchta, Inductive methods and zero-sum free sequences, arXiv:0711.1262 (2007). https://arxiv.org/abs/0711.1262
3. Weidong Gao, On Davenport's constant of finite abelian groups with rank three, Discrete Mathematics 222 (2000) 111–124, DOI 10.1016/S0012-365X(00)00010-8. https://doi.org/10.1016/S0012-365X(00)00010-8
4. Benjamin Girard and Wolfgang A. Schmid, Direct zero-sum problems for certain groups of rank three, Journal of Number Theory 197 (2019) 297–316, DOI 10.1016/j.jnt.2018.08.016 (preprint arXiv:1806.07636). https://doi.org/10.1016/j.jnt.2018.08.016
5. Maciej Zakarczemny, Note on the Davenport's constant for finite abelian groups with rank three, arXiv:1910.10984 (2019). https://arxiv.org/abs/1910.10984
6. Anamitro Biswas and Eshita Mazumdar, Davenport constant for finite abelian groups with higher rank, arXiv:2402.09999 (2024). https://arxiv.org/abs/2402.09999
7. Jun Seok Oh, A classification of finite groups with small Davenport constant, arXiv:2409.00363 (2024). https://arxiv.org/abs/2409.00363
8. John E. Olson, A combinatorial problem on finite Abelian groups, I, Journal of Number Theory 1 (1969) 8–10, DOI 10.1016/0022-314X(69)90021-3. https://doi.org/10.1016/0022-314X(69)90021-3
9. John E. Olson, A combinatorial problem on finite Abelian groups, II, Journal of Number Theory 1 (1969) 195–199, DOI 10.1016/0022-314X(69)90037-7. https://doi.org/10.1016/0022-314X(69)90037-7
10. Alfred Geroldinger and Rudolf Schneider, On Davenport's constant, Journal of Combinatorial Theory, Series A 61 (1992) 147–152, DOI 10.1016/0097-3165(92)90061-X. https://doi.org/10.1016/0097-3165(92)90061-X
11. Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin, and L. Rackham, Zero-sum problems in finite abelian groups and affine caps, The Quarterly Journal of Mathematics 58 (2007) 159–186, DOI 10.1093/qmath/ham003. https://doi.org/10.1093/qmath/ham003
12. Weidong Gao and Alfred Geroldinger, Zero-sum problems in finite abelian groups: a survey, Expositiones Mathematicae 24 (2006) 337–369, DOI 10.1016/j.exmath.2006.07.002. https://doi.org/10.1016/j.exmath.2006.07.002
13. P. van Emde Boas and D. Kruyswijk, A combinatorial problem on finite abelian groups III, Rapport ZW-1969-008, Mathematisch Centrum Amsterdam (1969), Zbl 0245.20046. https://zbmath.org/3387622
14. Charles Delorme, Oscar Ordaz, and Domingo Quiroz, Some remarks on Davenport constant, Discrete Mathematics 237 (2001) 119–128, DOI 10.1016/S0012-365X(00)00365-4. https://doi.org/10.1016/S0012-365X(00)00365-4