Restricted inverse zero-sum problems in groups of rank two

TL;DR

This paper investigates the structure of sequences in rank-two finite abelian groups that lack zero-sum subsequences of certain lengths, revealing complex structures and providing characterizations under conjectural and partial conditions.

Contribution

It offers a detailed analysis of inverse zero-sum problems in rank-two groups, including a complete characterization under a key conjecture and unconditional results in special cases.

Findings

Sequences without zero-sum subsequences have richer structures than expected.

Conditional complete characterization for groups of the form C_m ⊕ C_m.

Unconditional results obtained for specific cases.

Abstract

Let $(G,+)$ be a finite abelian group. Then, $\so(G)$ and $\eta(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to and at most, resp., the exponent of the group. For groups of rank two, we study the inverse problems associated to these constants, i.e., we investigate the structure of sequences of length $\so(G)-1$ and $\eta(G)-1$ that do not have such a subsequence. On the one hand, we show that the structure of these sequences is in general richer than expected. On the other hand, assuming a well-supported conjecture on this problem for groups of the form $C_m \oplus C_m$, we give a complete characterization of all these sequences for general finite abelian groups of rank two. In combination with partial results towards this conjecture, we get unconditional characterizations in special cases.