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The Erdős--Ginzburg--Ziv constant of Cnr for even n: rigid sequences, products, and growth beyond Harborth's bound

  • Christian Elsholtz
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Abstract

Let s(Cnr) be the Erdős--Ginzburg--Ziv constant of Cnr and η(Cnr) the short zero-sum constant. For even n, Harborth's construction gives s(Cnr)≥2r(n−1)+1, with equality when n is a power of two. Wang and Zeng recently showed that this bound is strict for every r≥4 and every even n that is not a power of two, with an excess of 2r−3(m−1), where m is the odd part of n; this disproves Properties C and D. We show that the excess grows faster than 2r. For every even n=2am with odd m≥3 and every r≥4, s(Cnr)≥2r(n−1)+(m−1) 2r−3⌊ r/4⌋+1 , so that s(Cnr)−2r(n−1)≥ cn r 2r.

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Cite

BibTeXRIS
BibTeX
@misc{elsholtz2026erdősginzburgziv,
  title = {{The Erdős--Ginzburg--Ziv constant of $C_n^r$ for even $n$: rigid sequences, products, and growth beyond Harborth's bound}},
  author = {Christian Elsholtz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35819},
  eprint = {2610.35819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35819},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Erdős--Ginzburg--Ziv constant of C_n^r for even n: rigid sequences, products, and growth beyond Harborth's bound
AU  - Christian Elsholtz
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.35819
UR  - https://arxiv.org/abs/2610.35819
AB  - Let s(C_n^r) be the Erdős--Ginzburg--Ziv constant of C_n^r and η(C_n^r) the short zero-sum constant. For even n, Harborth's construction gives s(C_n^r)≥2^r(n−1)+1, with equality when n is a power of two. Wang and Zeng recently showed that this bound is strict for every r≥4 and every even n that is not a power of two, with an excess of 2^r−3(m−1), where m is the odd part of n; this disproves Properties C and D. We show that the excess grows faster than 2^r. For every even n=2^am with odd m≥3 and every r≥4, s(C_n^r)≥2^r(n−1)+(m−1) 2^r−3⌊ r/4⌋+1 , so that s(C_n^r)−2^r(n−1)≥ c_n r 2^r.
ER  -