The $k$-th Davenport constants $\mathsf{D}_k(G)$ and their inverse problems for finite abelian groups of rank at least $3$

For a finite abelian group $G$ and a positive integer $k$, the $k$-th Davenport constant $\mathsf{D}_k(G)$ is the smallest length guaranteeing that every sequence over $G$ of at least that length contains $k$ disjoint nontrivial zero-sum subsequences. While $\mathsf{D}_k(G)$ is exactly known for all $k$ for cyclic groups, groups of rank at most $2$ (where $\mathsf{D}_k(G)=\mathsf{D}_0(G)+k\exp(G)$ with a single constant $\mathsf{D}_0(G)$ for every $k$), and elementary $2$-groups of rank at most $5$ and the elementary $3$-group $C_3^3$ of rank $3$, and while the sequence $(\mathsf{D}_k(G))_{k}$ is eventually an arithmetic progression with difference $\exp(G)$ for every $G$, the exact values and the extremal structure are open for essentially all groups of rank at least $3$: the all-$k$ linear pattern already fails for elementary $2$- and $3$-groups of rank at least $3$, and even bounding $\mathsf{D}_k(G)$ is considered substantially harder than for the classical Davenport constant. The problem, left open in Qinghai Zhong's 2025 resolution of the rank-$2$ inverse problem, is to determine $\mathsf{D}_k(G)$ exactly and to characterize the associated extremal class $\mathcal{M}_k(G)$ of maximal-length zero-sum sequences that cannot be partitioned into $k+1$ nontrivial zero-sum subsequences, for finite abelian groups of rank at least $3$, beyond the settled special cases.

Background

By a sequence over a finite abelian group $G$ (written additively) we mean an unordered finite list of elements of $G$ with repetition allowed. A sequence is a zero-sum sequence if the sum of its terms is $0$, and a zero-sum subsequence is a nonempty sub-multiset whose terms sum to $0$. The Davenport constant $\mathsf{D}(G)$ is the smallest $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nontrivial zero-sum subsequence; equivalently $\mathsf{D}(G)$ is the maximal length of a minimal zero-sum sequence, one that cannot be split into two nonempty zero-sum subsequences. Writing $G \cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $C_n$ the cyclic group of order $n$ and $n_1\mid\cdots\mid n_r$, the rank of $G$ is $r$ and the exponent $\exp(G)=n_r$ is the maximal order of an element; $G$ is an elementary $p$-group when $G\cong C_p^r$. The lower bound $\mathsf{D}(G)\ge 1+\sum_i(n_i-1)$ always holds, with equality for $p$-groups and rank at most $2$, but equality in rank $3$ is a famous open problem in general.

The $k$-th Davenport constant generalizes this: $\mathsf{D}_k(G)$ is the smallest $\ell$ such that every sequence over $G$ of length at least $\ell$ has $k$ disjoint nontrivial zero-sum subsequences, so $\mathsf{D}_1(G)=\mathsf{D}(G)$. This invariant was introduced by Halter-Koch (1992) in the arithmetic of non-unique factorizations, and it coincides, for abelian $G$, with the generalized Noether number $\beta_k(G)$ from invariant theory studied by Cziszter and Domokos. Knowledge of $\mathsf{D}_k(G)$ is also the key input to the inductive method, the main general technique for computing $\mathsf{D}(G)$ of groups built from quotients and subgroups, as exploited by Delorme, Ordaz and Quiroz (2001) and Bhowmik and Schlage-Puchta (2007).

Three facts delineate what is known. First, for rank at most $2$ one has the exact formula $\mathsf{D}_k(C_{n_1}\oplus C_{n_2})=n_1+kn_2-1$ (Geroldinger–Halter-Koch, Theorem 6.1.5), i.e. $\mathsf{D}_k(G)=\mathsf{D}_0(G)+k\exp(G)$ with the single constant $\mathsf{D}_0(G)=n_1-1$ for all $k$. Second, Freeze and Schmid (2010) proved that for every finite abelian group the sequence $(\mathsf{D}_k(G))_{k\ge1}$ is eventually an arithmetic progression with difference $\exp(G)$, so $\mathsf{D}_k(G)=\mathsf{D}_0(G)+k\exp(G)$ for all $k\ge k_D(G)$ for suitable $\mathsf{D}_0(G)$ and threshold $k_D(G)$; the proof is non-constructive and gives no effective general bound on $k_D(G)$. Third, the all-$k$ linear pattern fails in higher rank: elementary $2$- and $3$-groups of rank at least $3$ violate it (Delorme–Ordaz–Quiroz; Bhowmik–Schlage-Puchta). For elementary $2$-groups the exact values are known for rank at most $3$ (Delorme–Ordaz–Quiroz) and rank $4$ and $5$ (Freeze–Schmid, e.g. $\mathsf{D}_0(C_2^4)=5$, $k_D(C_2^4)=3$ with $\mathsf{D}_2(C_2^4)=8<5+2\cdot2$, and all $\mathsf{D}_k(C_2^5)$ with $\mathsf{D}_0(C_2^5)=11$, $k_D(C_2^5)=10$), while for rank $r\ge6$ even $\mathsf{D}_2(C_2^r)$ is known only within bounds of order $r$. Beyond the elementary $2$-groups, the elementary $3$-group $C_3^3$ of rank $3$ also has exact direct values, $\mathsf{D}_0(C_3^3)=6$, $k_D(C_3^3)=3$, $\mathsf{D}_1(C_3^3)=7$, $\mathsf{D}_2(C_3^3)=11$ (whence $\mathsf{D}_k(C_3^3)=3k+6$ for $k\ge3$), recorded in Freeze–Schmid (Remark 5.3); its inverse classification $\mathcal{M}_k(C_3^3)$ is open.

Zero-sum theory also studies inverse problems: characterizing the structure of extremal sequences, here the class $\mathcal{M}_k(G)$ of zero-sum sequences of length exactly $\mathsf{D}_k(G)$ that cannot be partitioned into $k+1$ nontrivial zero-sum subsequences. The classical inverse problem for $\mathsf{D}(G)$ was settled for rank $2$ in a series of works, and Zhong (2025) settled the inverse problem for $\mathsf{D}_k$ completely for rank $2$, characterizing $\mathcal{M}_k(C_n\oplus C_n)$ and $\mathcal{M}_k(C_{n_1}\oplus C_{n_2})$ by explicit basis normal forms. That paper, whose introduction states that computing or even sharply bounding $\mathsf{D}_k(G)$ outside rank $2$ is substantially more complicated, in particular for elementary $p$-groups, leaves the higher-rank direct and inverse problems open.

Problem Statement

For finite abelian groups $G$ of rank at least $3$, determine the exact values of the $k$-th Davenport constants $\mathsf{D}_k(G)$ for all positive integers $k$ — equivalently, determine the eventual-linear data $\mathsf{D}_0(G)$ and threshold $k_D(G)$ together with every pre-threshold value — and solve the associated inverse problem: characterize the extremal class $\mathcal{M}_k(G)$ of zero-sum sequences over $G$ of length exactly $\mathsf{D}_k(G)$ that cannot be partitioned into $k+1$ nontrivial zero-sum subsequences. The settled cases excluded from this question are: cyclic groups and groups of rank at most $2$ (where $\mathsf{D}_k(G)=\mathsf{D}_0(G)+k\exp(G)$ for all $k$ and $\mathcal{M}_k(G)$ is fully classified), the eventual-linearity theorem of Freeze–Schmid (which fixes only the asymptotic shape), and the exact direct values for elementary $2$-groups of rank at most $5$ and for the elementary $3$-group $C_3^3$ of rank $3$ (in these last cases $\mathsf{D}_k(G)$ is known exactly for every $k$, but the corresponding inverse classification $\mathcal{M}_k(G)$ is not). The following sub-questions are each independently open: (i) The $k=1$ instance, which includes as a subcase the still-open rank-$3$ Davenport constant conjecture $\mathsf{D}(G)=1+\sum_i(n_i-1)$. (ii) The direct constants and the inverse classification for elementary $2$-groups $\mathsf{D}_k(C_2^r)$ with $r\ge6$. (iii) The inverse classification $\mathcal{M}_k(C_3^3)$ for the elementary $3$-group of rank $3$, whose direct values are already known.

References

1. Qinghai Zhong, "On the inverse problem of the $k$-th Davenport constants for groups of rank $2$", arXiv:2503.21231 (2025); journal version DOI 10.1007/s00493-025-00153-3. URL: https://arxiv.org/abs/2503.21231
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