Sharp uniform bounds for the Davenport constant of rank-three finite abelian groups: Zakarczemny's Conjecture $D(G) \le D^*(G)+5(n_1-1)$, improving the explicit constant $a_3$, and the exact determination of $D(G)$

For a rank-three finite abelian group $G = C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$ with $1<n_1|n_2|n_3$, the Davenport constant $D(G)$ — the smallest length forcing a nonempty zero-sum subsequence in any sequence over $G$ — is not known exactly, and the best uniform upper bound has the form $D(G) \le D^*(G)+(a_3-3)(n_1-1)$, where $D^*(G)=n_1+n_2+n_3-2$ and $a_3$ is the optimal constant controlling the short zero-sum constant $\eta(C_n^3)$. Zakarczemny (2019/2021) proved $a_3 \le 20369$ and conjectured (Conjecture 3.12, motivated by a conjecture of Gao and Thangadurai, under which one may surmise the conjectural value $a_3=8$) that $D^*(G) \le D(G) \le D^*(G)+5(n_1-1)$ — a bound which follows from $a_3 \le 8$ via the quotient-group induction described below; only $a_3 \ge 7$ is known. As of 2026, no improvement of $a_3$, no resolution of the conjecture, and no exact determination of $D(G)$ for general rank-three groups is known. The problem — proving or refuting Conjecture 3.12, improving the explicit constant $a_3 \le 20369$ that controls the best uniform bound of this form, and, at full strength, exactly determining $D(G)$ for general rank-three groups — remains open.

Background

A sequence over a finite abelian group $G$ (written additively) is a collection of elements of $G$ in which repetition is allowed; a zero-sum subsequence is a nonempty sub-collection whose elements add up to $0$. The Davenport constant $D(G)$ is the smallest integer $t$ such that every sequence over $G$ of length at least $t$ contains a nonempty zero-sum subsequence; equivalently, one more than the maximal length of a zero-sumfree sequence (a sequence with no nonempty zero-sum subsequence). It is a central invariant of zero-sum theory, with applications to factorization theory, additive combinatorics, and lattice-point problems. Every finite abelian group decomposes uniquely as $G \cong C_{n_1}\oplus C_{n_2}\oplus\cdots\oplus C_{n_r}$ with cyclic summands of orders $1<n_1|n_2|\cdots|n_r$; $r$ is the rank of $G$ and $n_r=\exp(G)$ its exponent. An elementary counting argument gives the lower bound $D^*(G)=1+\sum_i (n_i-1) \le D(G)$. Classical theorems of Olson (1969) establish $D(G)=D^*(G)$ for all $p$-groups (finite groups whose order is a power of the prime $p$; every rank) and for all groups of rank at most two. Geroldinger and Schneider (1992) constructed infinitely many groups of ranks four and five with $D(G)>D^*(G)$, so the equality question — and more generally the exact determination of $D(G)$ — is open precisely in rank three, where it is a well-studied problem that has resisted decades of work.

Two relatives of $D(G)$ control the rank-three bounds. The short zero-sum constant $\eta(G)$ is the smallest $t$ such that every sequence over $G$ of length at least $t$ contains a nonempty zero-sum subsequence of length at most $\exp(G)$; the Erdős–Ginzburg–Ziv constant $\mathsf{s}(G)$ is the analogous constant for zero-sum subsequences of length exactly $\exp(G)$. Define $a_r$ as the optimal constant with $\eta(C_n^r) \le a_r(n-1)+1$ for all $n$; one knows $a_1=1$, $a_2=3$, and (by constructions due to Girard) $2^r-1 \le a_r$, so $7 \le a_3$. A standard quotient-group induction (for $G=C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$, quotient by a subgroup $H \cong C_{n_2/n_1}\oplus C_{n_3/n_1}$ so that $G/H \cong C_{n_1}^3$, using $D(G) \le D_{D(H)}(G/H)$, where $D_k(G)$ — the $k$-th higher Davenport constant — is the smallest length forcing $k$ pairwise disjoint nonempty zero-sum subsequences, together with $D_k(G) \le \exp(G)(k-1)+\eta(G)$) then yields $D(G) \le D^*(G)+(a_3-3)(n_1-1)$, so the resulting uniform bound on how far $D(G)$ can exceed $D^*(G)$ on rank-three groups, uniformly in the invariant factors, is governed by $a_3$.

The best known value of $a_3$ comes from lattice-point machinery: Alon and Dubiner (1995) proved $\mathsf{s}(C_p^r) \le c(r)p$ for primes $p$ with an explicit recursive constant, $c(3)<20233.005$; Edel, Elsholtz, Geroldinger, Kubertin and Rackham (2007) showed how such prime-case bounds on $\mathsf{s}$ lift to all composite $n$; and Chintamani, Moriya, Gao, Paul and Thangadurai (2012) and, in the $a_3$ formulation, Zakarczemny (2019; published 2021) combined these tools with the induction above to get $D(G) \le D^*(G)+20366(n_1-1)$, i.e. $a_3 \le 20369$ — more than three orders of magnitude above the conjectured value. On the asymptotic side, Girard (2018) proved $D(C_n^r) \sim rn$ as $n\to\infty$, so the gap $D-D^*$ is $o(n)$; but this result yields no effective uniform constant of the $a_r$ type and no exact values. On the conjectural side, Gao and Thangadurai (2006) studied the constants $\mathsf{s}_k(G)$ guaranteeing zero-sum subsequences of prescribed length $k\exp(G)$, and their Conjecture 0 on the rank-three group $C_n^3$ leads (as recorded by Zakarczemny) to the expectation $a_3=8$; Gao's conjecture $\eta(G)+\exp(G)-1=\mathsf{s}(G)$, true in rank at most two, connects $a_3$ to the still-open exact value of the Erdős–Ginzburg–Ziv constant of $C_n^3$. Zakarczemny's Conjecture 3.12 asserts the resulting uniform bound $D^*(G) \le D(G) \le D^*(G)+5(n_1-1)$ for all rank-three $G$. Exact values of $D(G)$ are known only for special families, e.g. the groups $C_2\oplus C_{n_2}\oplus C_{n_3}$ with $2|n_2|n_3$ settled by Girard and Schmid (2019).

Problem Statement

Sharpen the uniform upper bound on how far $D(G)$ can exceed $D^*(G)$ on rank-three groups $G=C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$, where $1<n_1|n_2|n_3$ and $D^*(G)=n_1+n_2+n_3-2$. The optimal constant $a_3$ in $\eta(C_n^3) \le a_3(n-1)+1$ controls, via the quotient-group induction of the Background, the best uniform bound of the form $D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3}) \le D^*(G)+(a_3-3)(n_1-1)$ obtainable through that induction; $a_3$ itself is not recovered from the values of $D(G)$, so the levels below are formulated directly in terms of the $D$-bounds. The problem has three sharpness levels of this one objective — pinning down how far $D(G)$ can exceed $D^*(G)$ on rank-three groups. Level (1) is Conjecture 3.12 of Zakarczemny, the exact-value question of level (3) is open, and level (2) is a weaker explicit-improvement direction relative to the proved bound $a_3 \le 20369$:

(1) Prove or refute Conjecture 3.12 of Zakarczemny: for every rank-three group $G=C_{n_1}\oplus C_{n_2}\oplus C_{n_3}$ with $1<n_1|n_2|n_3$, $D^*(G) \le D(G) \le D^*(G)+5(n_1-1)$ — a bound which follows from $a_3 \le 8$ via the quotient-group induction; since $a_3 \ge 7$ is known, the conjectured optimal value is $a_3=8$ (the expectation derived from the conjecture of Gao and Thangadurai).

(2) Independently of (1), improve the known explicit constant: establish a fully proved uniform bound $D(G) \le D^*(G)+(C-3)(n_1-1)$ valid for all rank-three groups, for some explicit $C<20369$; in particular, proving $a_3 \le C$ achieves this.

(3) The overarching, strongest form is the exact determination of $D(G)$ for general rank-three groups — in particular, deciding whether every rank-three group satisfies $D(G)=D^*(G)$, the last open rank of the classical Olson equality question.

These levels are related but not strictly nested: a proof of $a_3 \le 8$ would settle (1) and give (2) with $C=8$, and an exact determination at level (3) would settle (1); however, refuting Conjecture 3.12 by an explicit witness would resolve (1) without yielding any proved explicit $C<20369$, which is why (2) is stated independently of (1). The three levels are independent targets, and level (3) is the strongest form of the objective.

References

1. Maciej Zakarczemny, Note on the Davenport's constant for finite abelian groups with rank three, arXiv:1910.10984 (2019); published as Note on the Davenport constant for finite abelian groups with rank three, Acta Mathematica Universitatis Comenianae (N.S.) 90(1) (2021) 1-6, Zbl 1485.11145. DOI: 10.48550/arXiv.1910.10984, https://arxiv.org/abs/1910.10984
2. 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
3. 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
4. Alfred Geroldinger, Rudolf Schneider, On Davenport's constant, Journal of Combinatorial Theory, Series A 61(1) (1992) 147-152. DOI: 10.1016/0097-3165(92)90061-X, https://doi.org/10.1016/0097-3165(92)90061-X
5. Noga Alon, Moshe Dubiner, A lattice point problem and additive number theory, Combinatorica 15 (1995) 301-309. DOI: 10.1007/BF01299737, https://doi.org/10.1007/BF01299737
6. Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin, L. Rackham, Zero-sum problems in finite abelian groups and affine caps, The Quarterly Journal of Mathematics 58(2) (2007) 159-186. DOI: 10.1093/qmath/ham003, https://doi.org/10.1093/qmath/ham003
7. Weidong Gao, Ravindranathan Thangadurai, On zero-sum sequences of prescribed length, Aequationes Mathematicae 72(3) (2006) 201-212. DOI: 10.1007/s00010-006-2841-y, https://doi.org/10.1007/s00010-006-2841-y
8. M. N. Chintamani, B. K. Moriya, W. D. Gao, P. Paul, R. Thangadurai, New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants, Archiv der Mathematik 98(2) (2012) 133-142. DOI: 10.1007/s00013-011-0345-z, https://doi.org/10.1007/s00013-011-0345-z
9. Benjamin Girard, An asymptotically tight bound for the Davenport constant, Journal de l'École polytechnique — Mathématiques 5 (2018) 605-611. DOI: 10.5802/jep.79, arXiv:1709.08033, https://arxiv.org/abs/1709.08033
10. Benjamin Girard, Wolfgang 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, https://doi.org/10.1016/j.jnt.2018.08.016; arXiv:1806.07636, https://arxiv.org/abs/1806.07636
11. Maciej Zakarczemny, On the zero-sum constant, the Davenport constant and their analogues (published as The zero-sum constant, the Davenport constant and their analogues), Technical Transactions (2020), e2020027, 1-14; arXiv:1905.07648 (2019). DOI: 10.37705/techtrans/e2020027, https://arxiv.org/abs/1905.07648
12. Anamitro Biswas, Eshita Mazumdar, Davenport constant for finite abelian groups with higher rank, Mathematical Notes 118 (2025) 23-34; arXiv:2402.09999 (2024). DOI: 10.1134/S0001434624605483, https://arxiv.org/abs/2402.09999
13. Weidong Gao, Siao Hong, Jiangtao Peng, On zero-sum subsequences of length k exp(G), II, Journal of Combinatorial Theory, Series A 187 (2022), Article 105563. DOI: 10.1016/j.jcta.2021.105563, https://doi.org/10.1016/j.jcta.2021.105563
14. Siao Hong, Kevin Zhao, On the zero-sum subsequences of modular restricted lengths, Discrete Mathematics 347 (2024), Article 113967. DOI: 10.1016/j.disc.2024.113967, https://doi.org/10.1016/j.disc.2024.113967