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