Disproof of a Conjectured Upper Bound for the Davenport Constant
Abstract
Let G= Cn1⊕⋯⊕ Cnr be a finite abelian group with 1<n1|⋯| nr, and let rr(G)=r denote its rank. The Davenport constant DD(G) is the least integer ℓ such that every sequence of ℓ elements of G contains a nonempty zero-sum subsequence, and DD*(G)=1+∑i=1r(ni−1) is its classical lower bound. A long-standing conjecture asserts that DD(G)≤DD*(G)+rr(G)−1. In this paper, we disprove this conjecture. More strongly, we prove that suprr(G)=r(DD(G)−DD*(G))=∞ for every fixed r≥5. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. More precisely, our construction gives δ(Cmr−4⊕ C3m3⊕ C6m)≥ m for all integers r≥5 and m≥2. Consequently, the least rank r0 for unbounded Davenport excess DD(G)−DD*(G) satisfies 3≤ r0≤5. 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)≤16/5rr(G)exp(G) for every nontrivial finite abelian group G. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives DD(G)≤exp(G)(1+log|G|/exp(G)) ≤exp(G)(1+(rr(G)−1)logexp(G)). Our bound removes the logarithmic factor logexp(G) from this classical estimate, and replaces it with the absolute constant 16/5.
Related
BibTeX
@misc{wang2026disproof,
title = {{Disproof of a Conjectured Upper Bound for the Davenport Constant}},
author = {Guoqing Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.29878},
eprint = {2609.29878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Disproof of a Conjectured Upper Bound for the Davenport Constant
AU - Guoqing Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.29878
UR - https://arxiv.org/abs/2609.29878
AB - Let G= C_n_1⊕⋯⊕ C_n_r be a finite abelian group with 1<n_1|⋯| n_r, and let rr(G)=r denote its rank. The Davenport constant DD(G) is the least integer ℓ such that every sequence of ℓ elements of G contains a nonempty zero-sum subsequence, and DD^*(G)=1+∑_i=1^r(n_i−1) is its classical lower bound. A long-standing conjecture asserts that DD(G)≤DD^*(G)+rr(G)−1. In this paper, we disprove this conjecture. More strongly, we prove that sup_rr(G)=r(DD(G)−DD^*(G))=∞ for every fixed r≥5. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. More precisely, our construction gives δ(C_m^r−4⊕ C_3m^3⊕ C_6m)≥ m for all integers r≥5 and m≥2. Consequently, the least rank r_0 for unbounded Davenport excess DD(G)−DD^*(G) satisfies 3≤ r_0≤5. 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)≤16/5rr(G)exp(G) for every nontrivial finite abelian group G. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives DD(G)≤exp(G)(1+log|G|/exp(G)) ≤exp(G)(1+(rr(G)−1)logexp(G)). Our bound removes the logarithmic factor logexp(G) from this classical estimate, and replaces it with the absolute constant 16/5.
ER -