Equality $\mathsf{D}(G)=\mathsf{D}^*(G)$ for the Davenport constant of finite abelian groups and growth of the gap $\mathsf{D}(G)-\mathsf{D}^*(G)$

For a finite abelian group $G$, the Davenport constant $\mathsf{D}(G)$ is the smallest length forcing a nonempty zero-sum subsequence, and it satisfies the elementary lower bound $\mathsf{D}^*(G)=1+\sum_i (n_i-1)$ for $G\simeq C_{n_1}\oplus\cdots\oplus C_{n_r}$. Equality holds for $p$-groups and for groups of rank at most two, but fails in general; the central open problems are to determine exactly which finite abelian groups satisfy $\mathsf{D}(G)=\mathsf{D}^*(G)$ — in particular whether a rank-three counterexample exists and whether $\mathsf{D}(C_n^r)=r(n-1)+1$ for all $n$ — and to determine how the gap $\mathsf{D}(G)-\mathsf{D}^*(G)$ grows over natural families such as $C_n^r\oplus C_{kn}$. The lineage runs from the 1969 first counterexample through Geroldinger–Schneider, Mazur, and Liu's non-dispersive-sequence lower bounds to the 2024–2025 work of Biswas–Mazumdar, and all cores remain open as of 2026.

Background

Let $G$ be a finite abelian group, written additively. By the fundamental structure theorem, $G$ is isomorphic to a direct sum $C_{n_1}\oplus\cdots\oplus C_{n_r}$ of cyclic groups with $1<n_1\mid\cdots\mid n_r$, where $C_n$ denotes the cyclic group of order $n$; the number $r$ is the rank of $G$ and $n_r$ its exponent. A sequence over $G$ is a finite string of elements of $G$ in which repetition is allowed, and a zero-sum subsequence is a nonempty subcollection (respecting multiplicities) whose entries sum to $0$. The Davenport constant $\mathsf{D}(G)$ is the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a nonempty zero-sum subsequence; equivalently, $\mathsf{D}(G)-1$ is the maximal length of a zero-sum-free sequence, one with no nonempty zero-sum subsequence. Taking $n_i-1$ copies of each generator of the cyclic factors gives a zero-sum-free sequence, yielding the elementary lower bound $\mathsf{D}^*(G)=1+\sum_{i=1}^r (n_i-1)\le\mathsf{D}(G)$. Since the 1960s it has been known that equality $\mathsf{D}(G)=\mathsf{D}^*(G)$ holds for $p$-groups (groups whose order is a power of a prime $p$) and for all groups of rank at most two; see the survey of Gao and Geroldinger. The general statement $\mathsf{D}(G)=\mathsf{D}^*(G)$ is false: the first counterexample, found in 1969 in Amsterdam Mathematisch Centrum reports and documented in the paper of Geroldinger and Schneider and in Liu's paper, is the group $C_2^{4k}\oplus C_{4k+2}$ ($k\ge 1$), for which $\mathsf{D}(G)\ge\mathsf{D}^*(G)+1$. Geroldinger and Schneider (1992) constructed infinitely many groups of ranks four and five with $\mathsf{D}(G)>\mathsf{D}^*(G)$, for example $C_m\oplus C_n\oplus C_n\oplus C_{2n}$ for odd $m\mid n$, and Mazur (1992) proved the first quantitative growth result: for $G=C_2^{r-1}\oplus C_{2k}$ with $k>1$ odd, the gap $\mathsf{D}(G)-\mathsf{D}^*(G)$ grows at least like $\log_2 r$ minus a constant depending on $k$. Liu (2020) introduced the study of non-dispersive sequences over $C_n^r$ (sequences all of whose nonempty zero-sum subsequences have the same length, a notion going back to a conjecture of Graham) and proved that the gap over $G=C_n^r\oplus C_{kn}$ grows at least logarithmically in $r$ for fixed $n$, that the gap can be positive with arbitrarily large exponent, and a general theorem: if the exponent $m$ of $G$ is fixed and not a prime power, then for every $N$ there is $\varepsilon>0$ such that $|G|/m^r<\varepsilon$ forces $\mathsf{D}(G)-\mathsf{D}^*(G)>N$. On the upper-bound side, Zakarczemny (2019) proved that rank-three groups satisfy a linear upper bound $\mathsf{D}(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})\le (n_1-1)+(n_2-1)+(n_3-1)+1+(a_3-3)(n_1-1)$ with an explicit constant $a_3\le 20369$, still far from $\mathsf{D}^*$. On the equality side, $\mathsf{D}(G)=\mathsf{D}^*(G)$ is known for additional scattered families: Girard and Schmid determined the Davenport constant (via the multiwise Davenport constants, of which the ordinary one is the first) exactly for rank-three groups $C_2\oplus C_{n_2}\oplus C_{n_3}$ with $2\mid n_2\mid n_3$, and Biswas and Mazumdar (2024–2025) proved equality for $G=C_p^{d-1}\oplus C_{pq}$ whenever $p\mid q$, conditionally on an explicit conjecture when $\gcd(p,q)=1$, recording along the way the known equality family $C_3\oplus C_3\oplus C_{3s}$. Despite this progress, the basic structure of the problem is unresolved: no group of rank three is known to violate equality, no group of the form $C_n^r$ with $n$ having at least two distinct prime factors is known to violate it, and no sharp asymptotics of the gap are known for any natural family.

Problem Statement

Determine the exact relationship between the Davenport constant $\mathsf{D}(G)$ of a finite abelian group and its standard lower bound $\mathsf{D}^*(G)=1+\sum_{i=1}^r (n_i-1)$, where $G\simeq C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$. This comprises two linked questions (see Liu 2020; Biswas–Mazumdar 2024–2025). (1) Classification — for which finite abelian groups does $\mathsf{D}(G)=\mathsf{D}^*(G)$ hold? This question contains independently refutable instances; the sharpest open ones, and the required targets of this part, are: (a) does there exist a group of rank three with $\mathsf{D}(G)>\mathsf{D}^*(G)$ (equivalently, is $\mathsf{D}(G)=\mathsf{D}^*(G)$ for every rank-three group)? and (b) is $\mathsf{D}(C_n^r)=r(n-1)+1$ for every $n$ that is not a prime power and every rank $r\ge 3$? The full classification of all finite abelian groups is the overarching goal rather than a separately mandatory demand, and each of (a) and (b) is an independently refutable instance that may be settled on its own. (2) Gap growth — determine the sharp asymptotic behaviour of the difference $\mathsf{D}(G)-\mathsf{D}^*(G)$ over the family $C_n^r\oplus C_{kn}$ with $n,k\ge 2$ fixed and $r\to\infty$, that is, matching upper and lower bounds in $r$, where the best known results are Liu's lower bounds (at least logarithmic growth in $r$ for fixed $n$) with no matching upper bounds. Equivalently, (a) asks whether some rank-three group admits a zero-sum-free sequence of length $\mathsf{D}^*(G)$.

References

1. Chao Liu, "On the lower bounds of Davenport constant", Journal of Combinatorial Theory, Series A 171 (2020), article 105162. DOI: 10.1016/j.jcta.2019.105162; arXiv:1810.08346. https://arxiv.org/abs/1810.08346
2. Anamitro Biswas and Eshita Mazumdar, "Davenport Constant for Finite Abelian Groups with Higher Rank", Mathematical Notes 118 (2025), 23–34. DOI: 10.1134/S0001434624605483; arXiv:2402.09999. https://arxiv.org/abs/2402.09999
3. Maciej Zakarczemny, "Note on the Davenport's constant for finite abelian groups with rank three" (2019). arXiv:1910.10984. https://arxiv.org/abs/1910.10984
4. 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
5. Marcin Mazur, "A note on the growth of Davenport's constant", Manuscripta Mathematica 74 (1992), 229–235. DOI: 10.1007/BF02567669. https://doi.org/10.1007/BF02567669
6. 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
7. 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; arXiv:1806.07636. https://arxiv.org/abs/1806.07636
8. Benjamin Girard and Wolfgang A. Schmid, "Inverse zero-sum problems for certain groups of rank three", Acta Mathematica Hungarica 160 (2019), 229–247. DOI: 10.1007/s10474-019-00983-w. https://doi.org/10.1007/s10474-019-00983-w
9. 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