Zero-sum subsequences of length $k\exp(G)$ over finite abelian groups: the Gao–Han–Peng–Sun conjecture and the remaining rank-three cases

Let $G$ be a finite abelian group with exponent $\exp(G)$ and Davenport constant $\mathrm{D}(G)$, and let $\mathsf{s}_{k\exp(G)}(G)$ denote the least $\ell$ such that every sequence of $\ell$ elements of $G$ contains a zero-sum subsequence of length exactly $k\exp(G)$. A general lower bound $\mathsf{s}_{k\exp(G)}(G)\ge k\exp(G)+\mathrm{D}(G)-1$ holds, and a conjecture of Gao–Han–Peng–Sun (2014) asserts that equality holds whenever $k\exp(G)\ge\mathrm{D}(G)$. The conjecture is known for abelian $p$-groups with $p\ge5$ and $\mathrm{D}(G)\le4\exp(G)$, for elementary abelian $2$-groups, and for the rank-three groups $C_{2^n}^3$ and $C_{3^n}^3$ in most regimes of $k$; the outstanding rank-three cases are $\mathsf{s}_{3\cdot2^n}(C_{2^n}^3)$ and $\mathsf{s}_{4\cdot3^n}(C_{3^n}^3)$ for all $n\ge2$, conjecturally $6\cdot2^n-3$ and $7\cdot3^n-3$. The conjecture remains open as of August 2026. The related identity $\mathsf{s}_{k\exp(G)}(G)=\mathsf{s}_{\le k\exp(G)}(G)+k\exp(G)-1$, proposed in the literature, is in fact false in general (elementary abelian $2$-groups).

Background

Zero-sum theory studies when a sequence over a finite abelian group $G$ — a multiset of elements of $G$, repetitions allowed and order irrelevant — is forced to contain a sub-multiset whose terms sum to the identity element $0$ (a zero-sum subsequence). The field grew out of the 1961 theorem of Erdős, Ginzburg and Ziv that every sequence of $2n-1$ integers contains $n$ terms whose sum is divisible by $n$. Two invariants quantify this phenomenon. The Davenport constant $\mathrm{D}(G)$ is the least $\ell$ such that every sequence of $\ell$ elements of $G$ has a nonempty zero-sum subsequence; Olson (1969) proved $\mathrm{D}(G)=\sum_i(n_i-1)+1$ for $p$-groups (finite abelian groups in which every element has order a power of the prime $p$) and for groups $C_{n_1}\oplus\cdots\oplus C_{n_r}$ of rank two — the rank being the minimal number of generators — but no closed form is known in general. Writing $\exp(G)$ for the exponent of $G$ (the least common multiple of the element orders), the $k$-th generalized Erdős–Ginzburg–Ziv constant $\mathsf{s}_{k\exp(G)}(G)$ is the least $\ell$ such that every sequence of $\ell$ elements of $G$ contains a zero-sum subsequence of length exactly $k\exp(G)$; $\mathsf{s}(G):=\mathsf{s}_{\exp(G)}(G)$ is the classical Erdős–Ginzburg–Ziv constant. A companion invariant $\mathsf{s}_{\le t}(G)$ is defined analogously with the zero-sum subsequence required only to have length at most $t$. A simple padding argument gives the lower bound $\mathsf{s}_{k\exp(G)}(G)\ge k\exp(G)+\mathrm{D}(G)-1$ for all $k$. Gao (1996) proved that equality holds in the very-long-target regime $k\exp(G)\ge|G|$ (generalizing his $n+\mathrm{D}(G)-1$ theorem), and Gao–Thangadurai (2006) showed the bound is strict when $k\exp(G)<\mathrm{D}(G)$, so the critical window is exactly $k\exp(G)\ge\mathrm{D}(G)$. In this window, Gao–Han–Peng–Sun (2014) conjectured that equality always holds; an elementary-abelian precursor of the conjecture is due to Gao (2003). The conjecture has been verified for abelian $p$-groups with $\mathrm{D}(G)\le4\exp(G)$ and $p\ge5$ (Grynkiewicz, 2023, later gave a new proof for rank three based on the Chevalley–Warning theorem, a classical counting principle asserting that a system of polynomial congruences over a finite field has a number of common solutions divisible by the field characteristic whenever the degrees of the polynomials sum to less than the number of variables), and Sidorenko (2020) proved it for elementary abelian $2$-groups — direct sums of copies of the cyclic group $C_2$ of order two, i.e. vector spaces over the two-element field — showing $\mathsf{s}_{2m}(C_2^d)=2m+d$ whenever $d<2m$; his work connects these constants to binary linear codes (linear subspaces of the space of binary $n$-tuples, a codeword's weight being its number of nonzero entries) having no codeword of a given forbidden weight. For the rank-three groups $C_{p^n}^3$ (direct sum of three cyclic groups of order $p^n$), Han–Zhang (2019) and Zhang (2023) verified the conjecture for most $k$: for $p=2$ all $k\ge4$ and, by Gao–Thangadurai, $n=1$; for $p=3$ all $k\ge6$ (Han–Zhang), $k=3$ and $k=5$ with all $n\ge1$ (Zhang), and $k=4$ with $n=1$ (Gao–Thangadurai). What remains are $p=2,k=3,n\ge2$ and $p=3,k=4,n\ge2$. A further conjecture in the same circle, going back to Gao and coauthors, linked the two invariant families by the identity $\mathsf{s}_{k\exp(G)}(G)=\mathsf{s}_{\le k\exp(G)}(G)+k\exp(G)-1$; its $k=1$ case $\mathsf{s}(G)=\mathsf{s}_{\le\exp(G)}(G)+\exp(G)-1$ remains a well-known open conjecture, but the unrestricted identity is now known to be false in general, since Sidorenko (2020) determined $\mathsf{s}_{2m}(C_2^{2m+1})=4m+5$ for odd $m$ while $\mathsf{s}_{\le2m}(C_2^{2m+1})=2m+3$ for $m\ge2$ (Wang–Zhao, 2017), and $4m+5>4m+2$ for $m\ge3$. The refutation lies entirely in the regime $k\exp(G)<\mathrm{D}(G)$; when $k\exp(G)\ge\mathrm{D}(G)$ one has $\mathsf{s}_{\le k\exp(G)}(G)=\mathrm{D}(G)$, so there the identity is equivalent to the main conjecture above.

Problem Statement

Prove or disprove the following conjecture of Gao–Han–Peng–Sun — with the notation of the background above: $\exp(G)$ is the exponent of $G$, $\mathrm{D}(G)$ its Davenport constant, and $\mathsf{s}_{t}(G)$ the least $\ell$ such that every sequence of $\ell$ elements of $G$ contains a zero-sum subsequence of length exactly $t$ — namely: for every finite abelian group $G$ and every positive integer $k$ with $k\exp(G)\ge\mathrm{D}(G)$,

$$\mathsf{s}_{k\exp(G)}(G)=k\exp(G)+\mathrm{D}(G)-1.$$

In particular, determine the exact values of $\mathsf{s}_{3\cdot2^n}(C_{2^n}^3)$ and $\mathsf{s}_{4\cdot3^n}(C_{3^n}^3)$ for every $n\ge2$, where $C_{p^n}^3$ denotes the direct sum of three copies of the cyclic group of order $p^n$; the conjecture predicts

$$\mathsf{s}_{3\cdot2^n}(C_{2^n}^3)=6\cdot2^n-3,\qquad\mathsf{s}_{4\cdot3^n}(C_{3^n}^3)=7\cdot3^n-3.$$

The matching lower bounds are known, so the task in these cases is purely the upper bound: show that every sequence of $6\cdot2^n-3$ elements of $C_{2^n}^3$ contains a zero-sum subsequence of length $3\cdot2^n$, and every sequence of $7\cdot3^n-3$ elements of $C_{3^n}^3$ contains a zero-sum subsequence of length $4\cdot3^n$. A disproof of the general conjecture must exhibit a finite abelian group $G$, an integer $k$ with $k\exp(G)\ge\mathrm{D}(G)$, and a sequence of $k\exp(G)+\mathrm{D}(G)-1$ elements of $G$ with no zero-sum subsequence of length $k\exp(G)$. The two rank-three cases are instances of the general conjecture, not a substitute for it: a proof of the conjecture settles them, but a disproof — a counterexample over some finite abelian group — need not, so the task is resolved by proving or disproving the general conjecture.

References

1. Shiwen Zhang, 'On some zero-sum invariants for abelian groups of rank three', Publicationes Mathematicae Debrecen 106 (2025), no. 1-2, 225-240. DOI: 10.5486/pmd.2025.9991; arXiv:2310.05458. URL: https://doi.org/10.5486/pmd.2025.9991
2. Weidong Gao, Dongchun Han, Jiangtao Peng, Fang Sun, 'On zero-sum subsequences of length k exp(G)', Journal of Combinatorial Theory, Series A 125 (2014), 240-253. DOI: 10.1016/j.jcta.2014.03.006. URL: https://doi.org/10.1016/j.jcta.2014.03.006
3. 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. URL: https://doi.org/10.1016/j.jcta.2021.105563
4. Weidong Gao, Ravindranathan Thangadurai, 'On zero-sum sequences of prescribed length', Aequationes Mathematicae 72 (2006), 201-212. DOI: 10.1007/s00010-006-2841-y. URL: https://doi.org/10.1007/s00010-006-2841-y
5. W. D. Gao, 'On zero-sum subsequences of restricted size II', Discrete Mathematics 271 (2003), 51-59. DOI: 10.1016/S0012-365X(03)00038-4. URL: https://doi.org/10.1016/S0012-365X(03)00038-4
6. W. D. Gao, 'A combinatorial problem on finite abelian groups', Journal of Number Theory 58 (1996), 100-103. DOI: 10.1006/jnth.1996.0067. URL: https://doi.org/10.1006/jnth.1996.0067
7. Dongchun Han, Hanbin Zhang, 'On generalized Erdős–Ginzburg–Ziv constants of C_n^r', Discrete Mathematics 342 (2019), 1117-1127. DOI: 10.1016/j.disc.2018.12.018. URL: https://doi.org/10.1016/j.disc.2018.12.018
8. Alexander Sidorenko, 'On generalized Erdős–Ginzburg–Ziv constants for Z_2^d', Journal of Combinatorial Theory, Series A 174 (2020), Article 105254. DOI: 10.1016/j.jcta.2020.105254; arXiv:1808.06555. URL: https://doi.org/10.1016/j.jcta.2020.105254
9. David J. Grynkiewicz, 'A generalization of the Chevalley–Warning and Ax–Katz theorems with a view towards combinatorial number theory', Combinatorica 43 (2023), 1179-1213. DOI: 10.1007/s00493-023-00057-0; arXiv:2208.12895. URL: https://doi.org/10.1007/s00493-023-00057-0
10. Kevin Zhao, 'On zero-sum subsequences in a finite abelian group of length not exceeding a given number', arXiv preprint arXiv:2506.21383 (June 2025). URL: https://arxiv.org/abs/2506.21383
11. Silke Kubertin, 'Zero-sums of length kq in Z_q^d', Acta Arithmetica 116 (2005), 145-152. DOI: 10.4064/aa116-2-3. URL: https://doi.org/10.4064/aa116-2-3
12. Chunlin Wang, Kevin Zhao, 'On zero-sum subsequences of length not exceeding a given number', Journal of Number Theory 176 (2017), 365-374. DOI: 10.1016/j.jnt.2016.12.019. URL: https://doi.org/10.1016/j.jnt.2016.12.019
13. Weidong Gao, Dongchun Han, Hanbin Zhang, 'The EGZ-constant and short zero-sum sequences over finite abelian groups', Journal of Number Theory 162 (2016), 601-613. DOI: 10.1016/j.jnt.2015.09.024. URL: https://doi.org/10.1016/j.jnt.2015.09.024
14. Paul Erdős, A. Ginzburg, A. Ziv, 'A theorem in additive number theory', Bulletin of the Research Council of Israel 10F (1961), 41-43. No DOI. URL: https://zbmath.org/3102822
15. 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. URL: https://doi.org/10.1016/0022-314X(69)90021-3
16. Yongke Qu, Guoqing Wang, Yuanlin Li, 'The equality between the Erdős–Ginzburg–Ziv constant and the short product-one constant for finite nonabelian groups', arXiv preprint arXiv:2608.15568 (August 2026). URL: https://arxiv.org/abs/2608.15568