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PreprintCC BY 4.0arXiv

Additive bases and the zero-sum invariant N(G) for groups Cp⊕ Cnp

  • Wenkai Yang
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Abstract

Let G be a finite abelian group, and let c0(G) be the least integer such that every regular sequence over G of at least this length is an additive basis. Let N(G) be the least integer l such that every weak-regular sequence S over G with |S|≥ l has a nonempty zero-sum subsequence T satisfying vg(T)=vg(S) for some g| S. We obtain a common upper bound for c0(Cq⊕ Cnq) for groups of odd order, where q is prime, and prove N(G)≥ r(nr−1)+1 when G=Cn1⊕⋯⊕ Cnr with 1<n1|⋯| nr. If n≥1 and p≥3 is the smallest prime divisor of np, we obtain N(Cp⊕ Cnp2)=2np2−1 and give upper bounds for N(Cp⊕ Cnp) when pnmid n. We also determine N(Cn) for two infinite families of cyclic groups arising in earlier work.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026additive,
  title = {{Additive bases and the zero-sum invariant $\mathsfN(G)$ for groups $C_p\oplus C_np$}},
  author = {Wenkai Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13103},
  eprint = {2610.13103},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13103},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Additive bases and the zero-sum invariant N(G) for groups C_p⊕ C_np
AU  - Wenkai Yang
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13103
UR  - https://arxiv.org/abs/2610.13103
AB  - Let G be a finite abelian group, and let c_0(G) be the least integer such that every regular sequence over G of at least this length is an additive basis. Let N(G) be the least integer l such that every weak-regular sequence S over G with |S|≥ l has a nonempty zero-sum subsequence T satisfying v_g(T)=v_g(S) for some g| S. We obtain a common upper bound for c_0(C_q⊕ C_nq) for groups of odd order, where q is prime, and prove N(G)≥ r(n_r−1)+1 when G=C_n_1⊕⋯⊕ C_n_r with 1<n_1|⋯| n_r. If n≥1 and p≥3 is the smallest prime divisor of np, we obtain N(C_p⊕ C_np^2)=2np^2−1 and give upper bounds for N(C_p⊕ C_np) when pnmid n. We also determine N(C_n) for two infinite families of cyclic groups arising in earlier work.
ER  -