2021/09/21 by David J. Grynkiewicz, Chao Liu, Grynkiewicz, David J. +1 · 1 citation
Computer Science · Mathematics · #11B75 #Analytic Number Theory Research #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2109.10300
openalex publication_date 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G=Cn⊕ Cn 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+1)/(3)] was recently established, showing S must have the form S=e1[n-1]\boldsymbol⋅e2[n -1]\boldsymbol⋅ (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+1)/(3), or when n is composite and k=n-d-1 or n-2d+1 for a proper, nontrivial divisor d| n.