A Multiplicative Property for Zero-Sums II

For $n\geq 1$, let $C_n$ denote a cyclic group of order $n$. Let $G=C_n\oplus C_{mn}$ with 2$ and $m\geq $k\in [0,n-1]$. It is known that any sequence $mn+n-1+k$ terms from $G$ must contain nontrivial zero-sum length at most $mn+n-1-k$. The associated inverse question to characterize those sequences maximal $mn+n-2+k$ fail subsequence $k\leq this the for Davenport Constant. $k=n-1$, $\eta(G)$ invariant concerning short subsequences. $C_n\oplus C_n$ [2,n-2]$, 5$ prime, it was conjectured in paper Grynkiewicz, Wang Zhao they have form $S=e_1^{[n-1]}\boldsymbol{\cdot} e_2^{[n-1]}\boldsymbol{\cdot} (e_1+e_2)^{[k]}$ some basis $(e_1,e_2)$, conjecture established many cases later extended composite moduli In paper, we focus on case 2$. Assuming structure holds [2,n-2]$ C_n$, all $mn+n-1-k$, showing either e_2^{[sn-1]}\boldsymbol{\cdot} (e_1+e_2)^{[(m-s)n+k]}$ $s\in [1,m]$ $(e_1,e_2)$ $\mathsf{ord}(e_2)=mn$, or else $S=g_1^{[n-1]}\boldsymbol{\cdot} g_2^{[n-1]}\boldsymbol{\cdot} (g_1+g_2)^{[(m-1)n+k]}$ generating set $\{g_1,g_2\}$ $\mathsf{ord}(g_1+g_2)=mn$. view prior work, reduces structural characterization general rank two abelian $C_p\oplus C_p$ $p$ prime. Additionally, give new proof precise $k=n-1$ $m=1$. Combined results, our results unconditionally establish extremal cases, including when $n$ only divisible by primes $7$, prime power \frac{2n+1}{3}$, $k=n-d-1$ $n-2d+1$ proper, divisor $d\mid n$.