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Davenport’s constant for groups of the form ${ℤ}_{3}\oplus {ℤ}_{3}\oplus {ℤ}_{3d}$. (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 $D\left(G\right)$ 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 $G={ℤ}_{{d}_{1}}\oplus \cdots \oplus {ℤ}_{{d}_{r}}$, where ${d}_{i}$ divides ${d}_{i+1}$. Then one easily obtains a lower bound for the Davenport constant: $D\left(G\right)\ge M\left(G\right):=1-r+{\sum }_{i}{d}_{i}$. For groups of rank $r\le 2$, one has equality. For groups of rank $\ge 4$, there are infinitely examples with $D\left(G\right)>M\left(G\right)$. Whether one always has equality in rank 3 is still open. The present article proves equality for the groups $G:={ℤ}_{3}\oplus {ℤ}_{3}\oplus {ℤ}_{3d}$, with $d\in ℕ$. (Before, the smallest unknown case was ${ℤ}_{3}\oplus {ℤ}_{3}\oplus {ℤ}_{15}$.)

The method used in the proof is an enhancement of the “inductive method” used by C. Delorme, O. Ordaz and D. Quiroz [Discrete Math. 237, No. 1–3, 119–128 (2001; Zbl 1003.20025)]. Using the projection $\pi :G\to {ℤ}_{3}^{3}$, each multiset $A\subset G$ yields a multiset $\stackrel{˜}{A}:=\pi \left(A\right)\subset {ℤ}_{3}^{3}$. 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 $G\cong {ℤ}_{3}^{3}\oplus {ℤ}_{d}$, and the information which gets lost by applying $\pi$ to $A$ can be encoded in a map $f:\stackrel{˜}{A}\to {ℤ}_{d}$. Using these notations, a subset $B\subset A$ has sum zero if and only if its projection $\pi \left(B\right)\subset \stackrel{˜}{A}$ has sum zero and ${\sum }_{b\in \pi \left(B\right)}f\left(b\right)=0$. In this way, the question about multisets in $G$ has been translated into a question about multisets in ${ℤ}_{3}^{3}$ with functions on them; this problem is easier to solve (but requires a lot of computation in ${ℤ}_{3}^{3}$).

MSC:
 11B75 Combinatorial number theory 20K01 Finite abelian groups 20D60 Arithmetic and combinatorial problems on finite groups