Davenport Constants of Finite Abelian Groups

Abstract

A central problem in zero-sum theory is to determine the Davenport constant of a finite abelian group. For a finite abelian group G≅ C_n₁⊕⋯⊕ C_{n_r}, with 1<n₁|⋯| n_r, let D(G) be the least integer t such that every sequence of t elements of G contains a nonempty zero-sum subsequence, and denote D^*(G):= 1+∑_i=1^r(n_i−1). The standard construction shows that D(G)≥D^*(G). Equality holds for several important classes of groups but fails in general, and even the question whether D(G)/D^*(G) is bounded by an absolute constant remains open. In this paper, we prove the upper bound D(G)<4.86 exp(G)log|G|/logexp(G) for every finite abelian group. When |G| is large compared with exp(G), it improves the previous classical upper bound by a factor of order logexp(G). We also prove that, for almost all finite abelian groups (counting isomorphism classes by order), D(G)∼exp(G)log|G|/logexp(G)∼D^*(G). We also study the difference D(G)−D^*(G): for every function L with L(x)→∞, almost all groups satisfy D(G)−D^*(G)≤ L(D^*(G)). However, for every fixed C≥0, the groups with D(G)−D^*(G)>C have positive lower density.

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BibTeX RIS

BibTeX

@misc{han2026davenport,
  title = {{Davenport Constants of Finite Abelian Groups}},
  author = {Dongchun Han and Hanbin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15102},
  eprint = {2610.15102},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15102},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Davenport Constants of Finite Abelian Groups
AU  - Dongchun Han
AU  - Hanbin Zhang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15102
UR  - https://arxiv.org/abs/2610.15102
AB  - A central problem in zero-sum theory is to determine the Davenport constant of a finite abelian group. For a finite abelian group G≅ C_n_1⊕⋯⊕ C_n_r, with 1<n_1|⋯| n_r, let D(G) be the least integer t such that every sequence of t elements of G contains a nonempty zero-sum subsequence, and denote D^*(G):= 1+∑_i=1^r(n_i−1). The standard construction shows that D(G)≥D^*(G). Equality holds for several important classes of groups but fails in general, and even the question whether D(G)/D^*(G) is bounded by an absolute constant remains open. In this paper, we prove the upper bound D(G)<4.86 exp(G)log|G|/logexp(G) for every finite abelian group. When |G| is large compared with exp(G), it improves the previous classical upper bound by a factor of order logexp(G). We also prove that, for almost all finite abelian groups (counting isomorphism classes by order), D(G)∼exp(G)log|G|/logexp(G)∼D^*(G). We also study the difference D(G)−D^*(G): for every function L with L(x)→∞, almost all groups satisfy D(G)−D^*(G)≤ L(D^*(G)). However, for every fixed C≥0, the groups with D(G)−D^*(G)>C have positive lower density.
ER  -

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