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Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three

  • Hanbin Zhang
  • Shiwen Zhang
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Abstract

Let G be a finite abelian group with exponent exp(G). For positive integer k, the generalized Erdős--Ginzburg--Ziv constant, denoted by skexp(G)(G) is the least integer ℓ such that every sequence of length at least ℓ over G has a zero-sum subsequence of length kexp(G). In this paper, we prove that, for any n≥ 1, s3· 2n(C2n3)=6· 2n−3 and s4· 3n(C3n3)=7· 3n−3. This completes all the remaining cases in the study of skq(Cq3), where q is a prime power and k≥ 3. As a consequence, for every k≥ 3, we have skn(Cn3)=(k+3)n+O(n/(log n)). This verifies a conjecture of Kubertin asymptotically. Our approach combines the previous group ring techniques and a recent powerful generalization of the Chevalley-Warning and Ax-Katz theorems introduced by Grynkiewicz

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026generalized,
  title = {{Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three}},
  author = {Hanbin Zhang and Shiwen Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21577},
  eprint = {2610.21577},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21577},
}
RIS
TY  - UNPB
TI  - Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three
AU  - Hanbin Zhang
AU  - Shiwen Zhang
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21577
UR  - https://arxiv.org/abs/2610.21577
AB  - Let G be a finite abelian group with exponent exp(G). For positive integer k, the generalized Erdős--Ginzburg--Ziv constant, denoted by s_kexp(G)(G) is the least integer ℓ such that every sequence of length at least ℓ over G has a zero-sum subsequence of length kexp(G). In this paper, we prove that, for any n≥ 1, s_3· 2^n(C_2^n^3)=6· 2^n−3 and s_4· 3^n(C_3^n^3)=7· 3^n−3. This completes all the remaining cases in the study of s_kq(C_q^3), where q is a prime power and k≥ 3. As a consequence, for every k≥ 3, we have s_kn(C_n^3)=(k+3)n+O(n/(log n)). This verifies a conjecture of Kubertin asymptotically. Our approach combines the previous group ring techniques and a recent powerful generalization of the Chevalley-Warning and Ax-Katz theorems introduced by Grynkiewicz
ER  -