A multiplicative property for zero-sums I

Let G=Cn×Cn, where Cn denotes a cyclic group of order n, and let k∈[0,n−1]. We study the structure sequences terms from G with maximal length |S|=2n−2+k that fail to contain nontrivial zero-sum subsequence at most 2n−1−k. For k≤1, this is inverse question for Davenport Constant. k=n−1, η(G) invariant concerning short subsequences. The in both these cases (known respectively as Property B C) was established two-step process: first verifying multiplicative property that, if structural description holds when n=n1 n=n2, then it n=n1n2, resolving case n prime separately. When prime, characterization k∈[2,2n+13] recently established, showing S must have form S=e1[n−1]⋅e2[n−1]⋅(e1+e2)[k] some basis (e1,e2) G. It conjectured also k∈[2,n−2] (when prime). In paper, we extend conjecture by dropping restriction be establish following result. Suppose k=kmn+kn km∈[0,m−1] kn∈[0,n−1]. If km Cm×Cm kn Cn×Cn, k Cmn×Cmn. This reduces full case. Combined known results, unconditionally establishes extremal G=Cn×Cn many cases, including only divisible primes 7, n≥2 power k≤2n+13, or composite k=n−d−1 n−2d+1 proper, divisor d|n.