Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 11 min ago
PreprintCC BY 4.0arXiv

The equality s(G)=η(G)+exp(G)−1 fails for G=C27⊕ C6

  • Guoqing Wang
  • Xiangneng Zeng
Open at sourcePDFDOICite

Abstract

Let G be a finite abelian group. The Erdős-Ginzburg-Ziv constant s(G) is the smallest positive integer ℓ such that every sequence over G of length at least ℓ contains a zero-sum subsequence of length exp(G). The short zero-sum constant η(G) is defined similarly, with the requirement that the zero-sum subsequence is nonempty and has length at most exp(G). It is known that s(G)≥ η(G)+exp(G)−1. A conjecture proposed by W. Gao in 2003, which has been widely studied in zero-sum theory, predicts that s(G)=η(G)+exp(G)−1 for every nontrivial finite abelian group G. In this paper, we prove that this conjecture fails for the group C27⊕ C6.

Related

  1. The Wehrl-type entropy conjecture for symmetric SU(N) coherent states: Cases of equality and stability
    Fabio Nicola, Federico Riccardi, Paolo Tilli
    Duke Mathematical JournalPeer-reviewedLicence not statedFull text11 citedMathematical physicsOpenAlex
  2. The silp--spli equality for group rings
    Xiaolei Zhang, Guocheng Dai
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. A new proof of equality cases of the Brunn-Minkowski inequality on manifolds
    Osama Farooqui
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  4. A characterization of equality between de Branges--Rovnyak and Dirichlet spaces
    Carlo Bellavita, Eugenio Dellepiane, Shuaibing Luo
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
  5. The case of equality in BK
    Raphaël Cerf, Pierre Tesio
    arXivPreprintCC BY 4.0Full textProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{wang2026equality,
  title = {{The equality $s(G)=η(G)+\exp(G)-1$ fails for $G=C_2^7\oplus C_6$}},
  author = {Guoqing Wang and Xiangneng Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01972},
  eprint = {2611.01972},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01972},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The equality s(G)=η(G)+exp(G)−1 fails for G=C_2^7⊕ C_6
AU  - Guoqing Wang
AU  - Xiangneng Zeng
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01972
UR  - https://arxiv.org/abs/2611.01972
AB  - Let G be a finite abelian group. The Erdős-Ginzburg-Ziv constant s(G) is the smallest positive integer ℓ such that every sequence over G of length at least ℓ contains a zero-sum subsequence of length exp(G). The short zero-sum constant η(G) is defined similarly, with the requirement that the zero-sum subsequence is nonempty and has length at most exp(G). It is known that s(G)≥ η(G)+exp(G)−1. A conjecture proposed by W. Gao in 2003, which has been widely studied in zero-sum theory, predicts that s(G)=η(G)+exp(G)−1 for every nontrivial finite abelian group G. In this paper, we prove that this conjecture fails for the group C_2^7⊕ C_6.
ER  -