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Davenport’s constant for groups of the form $\Bbb Z_3\oplus\Bbb Z_3\oplus\Bbb Z_{3d}$. (English) Zbl 1173.11012
Granville, Andrew (ed.) et al., Additive combinatorics. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4351-2/pbk). CRM Proceedings and Lecture Notes 43, 307-326 (2007).
Let $G$ be a finite abelian group. The Davenport constant $D(G)$ is the least integer $n$ such that any multiset consisting of $n$ elements of $G$ has a (non-empty) subset whose sum is $0$. Write $G = \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_r}$, where $d_i$ divides $d_{i+1}$. Then one easily obtains a lower bound for the Davenport constant: $D(G) \ge M(G) := 1 - r + \sum_i d_i$. For groups of rank $r \le 2$, one has equality. For groups of rank $\ge 4$, there are infinitely examples with $D(G) > M(G)$. Whether one always has equality in rank 3 is still open. The present article proves equality for the groups $G := \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{3d}$, with $d \in \mathbb{N}$. (Before, the smallest unknown case was $\mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{15}$.) The method used in the proof is an enhancement of the “inductive method” used by {\it C. Delorme, O. Ordaz} and {\it D. Quiroz} [Discrete Math. 237, No. 1--3, 119--128 (2001; Zbl 1003.20025)]. Using the projection $\pi: G \to \mathbb{Z}_{3}^3$, each multiset $A \subset G$ yields a multiset $\tilde{A} := \pi(A) \subset \mathbb{Z}_{3}^3$. Now suppose that $d$ is coprime to $3$. (The case $3\mid d$ has already been treated by Delorme, Ordaz and Quiroz.) Then we have $G \cong \mathbb{Z}_{3}^3 \oplus \mathbb{Z}_d$, and the information which gets lost by applying $\pi$ to $A$ can be encoded in a map $f: \tilde{A} \to \mathbb{Z}_d$. Using these notations, a subset $B \subset A$ has sum zero if and only if its projection $\pi(B) \subset \tilde{A}$ has sum zero and $\sum_{b \in \pi(B)} f(b) = 0$. In this way, the question about multisets in $G$ has been translated into a question about multisets in $\mathbb{Z}_{3}^3$ with functions on them; this problem is easier to solve (but requires a lot of computation in $\mathbb{Z}_{3}^3$). For the entire collection see [Zbl 1124.11003].

11B75Combinatorial number theory
20K01Finite abelian groups
20D60Arithmetic and combinatorial problems on finite groups
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