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Davenport’s constant for groups of the form 3 3 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= d 1 d r , where d i divides d i+1 . Then one easily obtains a lower bound for the Davenport constant: D(G)M(G):=1-r+ i d i . For groups of rank r2, one has equality. For groups of rank 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:= 3 3 3d , with d. (Before, the smallest unknown case was 3 3 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 π:G 3 3 , each multiset AG yields a multiset A ˜:=π(A) 3 3 . Now suppose that d is coprime to 3. (The case 3d has already been treated by Delorme, Ordaz and Quiroz.) Then we have G 3 3 d , and the information which gets lost by applying π to A can be encoded in a map f:A ˜ d . Using these notations, a subset BA has sum zero if and only if its projection π(B)A ˜ has sum zero and bπ(B) f(b)=0. In this way, the question about multisets in G has been translated into a question about multisets in 3 3 with functions on them; this problem is easier to solve (but requires a lot of computation in 3 3 ).

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