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Disproof of a Conjectured Upper Bound for the Davenport Constant — arXiv:2609.29878

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.29878
License (as archived): CC0-1.0
Archived: 2026-10-08T06:40:59.956Z
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Disproof of a Conjectured Upper Bound for the Davenport Constant — arXiv:2609.29878 — arXiv (en), https://arxiv.org/abs/2609.29878 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bf4bd0ea2653 (Metraipolis archive dc_bf4bd0ea2653, sha256 610f13325fdc0d34fd51b1647babd61e40a166cd9b8aaa4e7bd885801a497134)

BibTeX

@misc{dcbf4bd0ea2653,
  title = {Disproof of a Conjectured Upper Bound for the Davenport Constant — arXiv:2609.29878},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.29878},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bf4bd0ea2653; sha256 610f13325fdc0d34fd51b1647babd61e40a166cd9b8aaa4e7bd885801a497134},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.29878. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture \cite[Conjecture 3.7]{GG06} asserts that $\DD(G)\le\DD^*(G)+\rr(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{\rr(G)=r}\bigl(\DD(G)-\DD^*(G)\bigr)=\infty$ for every fixed $r\ge8$. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. Our result also disproves the Narkiewicz--Śliwa conjecture of 1982 \cite{NS82} on the Narkiewicz constant, arising in algebraic number theory from the quantitative study of algebraic integers with unique factorization. The same amplification of the Davenport excess yields counterexamples to Girard's conjecture \cite[Conjecture 1.2]{Girard08} on the cross numbers of long zero-sum-free sequences. As a further main result, we establish the uniform upper bound $$\DD(G)\le\frac{16}{5}\rr(G)\exp(G)$$ for every nontrivial finite abelian group $G$. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives $\DD(G)\le\exp(G)\left(1+\log\frac{|G|}{\exp(G)}\right) \le\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$. Our bound removes the logarithmic factor $\log\exp(G)$ from this classical estimate, and replaces it with the absolute constant $16/5$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-25 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Number Theory (math.NT).
Pagina del lavoro: https://arxiv.org/abs/2609.29878 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T06:02:10.074Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-25?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).