In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈N such that every sequence S over G of length ∣S∣≥l has a non-empty zero-sum subsequence.
[GG2006-def-D]
For n≥2, let Cn denote the cyclic group of order n, and write
Cn3 := Cn⊕Cn⊕Cn.
[GG2006-def-Cn]
We define
C53 := n≥2sup n−1D(Cn3)−1,
the maximal normalized Davenport constant among the rank-3 groups Cn3.
The following explicit uniform inequality is obtained from the inductive method
and known multi-wise Davenport estimates over elementary prime groups:
3(n−1)+1 ≤ D(Cn3) ≤ 4n−P(n)−2(n≥2),
where
P(n)=pa∥nmaxpa
is the largest primary component of n. Since P(n)≥2, this gives
D(Cn3)≤4n−4=4(n−1),
and hence
C53≤4.
The previously recorded published bounds included
3(n−1)+1 ≤ D(Cn3) ≤ min{20369, 3ω(n)}(n−1)+1,
where ω(n) is the number of distinct prime factors of n.
[Zak2019-omega-def] [Zak2019-cor3.11]
Here, the 3ω(n) term is inherited from earlier published work, while the uniform numeric constant 20369 is the explicit contribution in [Zak2019].
[Zak2019-prev-3omega] [CMMPT2012]
In particular,
3 ≤ C53 ≤ 4.
A long-standing conjecture (in a stronger, pointwise form) predicts that for all n≥2 one has
D(Cn3) = 3(n−1)+1,
equivalently C53=3.
[GG2006-conj3.5] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]
One unconditional family of exact evaluations is given by prime powers: if n is a prime power, then Cn3 is a p-group, and Theorem 3.1 implies d(Cn3)=d\*(Cn3)=3(n−1) and hence D(Cn3)=3(n−1)+1.
[GG2006-thm3.1] [GG2006-D-equals-1-plus-d] [GG2006-def-dstar]
The general determination of D(G) for rank-3 groups (and in particular the pointwise determination of D(Cn3) for all n) remains open.
[Zak2019-open-rank3]
Published work before the 2019 preprint includes Gao (2000) on rank-3 groups and Chintamani--Moriya--Gao--Paul--Thangadurai (2012), which gives the bound D(Cn3)≤3ω(n)(n−1)+1 used in Corollary 3.11.
[Zak2019-ref-Gao2000] [Zak2019-prev-3omega] [CMMPT2012] [Gao2000]