A Multiplicative Property for Zero-Sums II

Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to characterize those sequences with maximal length $mn+n-2+k$ that fail to contain a nontrivial zero-sum subsequence of length at most $mn+n-1-k$. For $k\leq 1$, this is the inverse question for the Davenport Constant. For $k=n-1$, this is the inverse question for the $\eta(G)$ invariant concerning short zero-sum subsequences. The structure in both these cases is known, and the structure for $k\in [2,n-2]$ when $m=1$ was studied previously with it conjectured that they must have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[n-1]}\boldsymbol{\cdot} (e_1+e_2)^{[k]}$ for some basis $(e_1,e_2)$, with the conjecture established in many cases. We focus on $m\geq 2$. Assuming the conjectured structure holds for $k\in [2,n-2]$ in $C_n\oplus C_n$, we characterize the structure of all sequences of maximal length $mn+n-2+k$ in $C_n\oplus C_{mn}$ that fail to contain a nontrivial zero-sum of length at most $mn+n-1-k$, showing they must have either have the form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[sn-1]}\boldsymbol{\cdot} (e_1+e_2)^{[(m-s)n+k]}$ for some $s\in [1,m]$ and basis $(e_1,e_2)$ with $\mathsf{ord}(e_2)=mn$, or else have the form $S=g_1^{[n-1]}\boldsymbol{\cdot} g_2^{[n-1]}\boldsymbol{\cdot} (g_1+g_2)^{[(m-1)n+k]}$ for some generating set $\{g_1,g_2\}$ with $\mathsf{ord}(g_1+g_2)=mn$. Additionally, we give a new proof of the precise structure in the case $k=n-1$ for $m=1$. Combined with known results, our results unconditionally establish the structure of extremal sequences in $G=C_n\oplus C_{mn}$ in many cases.