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PreprintCC BY 4.0arXiv

On a classical zero-sum invariant II: Disproof of a long-standing conjecture

  • Alfred Geroldinger
  • Guoqing Wang
  • Wenkai Yang
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Abstract

For a nontrivial finite abelian group G, let ν(G) be the smallest {nonnegative} integer ℓ such that every zero-sum free sequence T over G of length at least ℓ has the following property: all nonzero elements of G that do not occur as a subsequence sum of T lie in a proper coset of some subgroup of G. It is easy to check that d (G)−1 ≤ ν(G) ≤ d (G), where d (G) is the small Davenport constant of G. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In this paper, we disprove the conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{geroldinger2026classical,
  title = {{On a classical zero-sum invariant II: Disproof of a long-standing conjecture}},
  author = {Alfred Geroldinger and Guoqing Wang and Wenkai Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.17127},
  eprint = {2609.17127},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17127},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a classical zero-sum invariant II: Disproof of a long-standing conjecture
AU  - Alfred Geroldinger
AU  - Guoqing Wang
AU  - Wenkai Yang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.17127
UR  - https://arxiv.org/abs/2609.17127
AB  - For a nontrivial finite abelian group G, let ν(G) be the smallest {nonnegative} integer ℓ such that every zero-sum free sequence T over G of length at least ℓ has the following property: all nonzero elements of G that do not occur as a subsequence sum of T lie in a proper coset of some subgroup of G. It is easy to check that d (G)−1 ≤ ν(G) ≤ d (G), where d (G) is the small Davenport constant of G. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In this paper, we disprove the conjecture.
ER  -