Inverse Zero-Sum Problems III: Addendum

TL;DR

This paper corrects previous gaps in the characterization of maximal zero-sum sequences in the group (Z/nZ)^2, providing essential clarifications for the inverse zero-sum problem and supporting future comprehensive work.

Contribution

It identifies and rectifies missing cases in earlier proofs of the inverse zero-sum problem for (Z/nZ)^2, ensuring accurate characterization of maximal zero-sum sequences.

Findings

Corrected the proof of the inverse zero-sum problem for (Z/nZ)^2

Filled gaps in the classification of minimal zero-sum sequences of maximal length

Prepared the groundwork for a comprehensive future characterization

Abstract

The Davenport constant for a finite abelian group $G$ is the minimal length $\ell$ such that any sequence of $\ell$ terms from $G$ must contain a nontrivial zero-sum sequence. For the group $G=(\mathbb Z/n\mathbb Z)^2$, its value is $2n-1$, which is a classical result of Olson. The associated inverse question is to characterize those sequences of maximal length $2n-2$ that do not have a nontrivial zero-sum subsequence. A simple argument shows this to be equivalent to characterizing all minimal zero-sum sequences of maximal length $2n-1$, with a minimal zero-sum sequence being one that cannot have its terms partitioned into two proper, nontrivial zero-sum subsequences. This was done in a series of papers . However, there is a missing case in one of the required papers, leading to a missing case not treated in the portion of the proof. Both these deficiencies are corrected here. The correction contained in this manuscript will be incorporated into a larger, forthcoming work that encompasses the entirety of the characterization, and is posted in advance so that the corrected argument is made available to the research community before this larger project is completed.