Structure of a sequence with prescribed zero-sum subsequences: Rank Two p-groups

2022/11/15 by Ebert, John, Grynkiewicz, David J. · 1 citation
#11B75 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2211.08515

Abstract

Let G=(\mathbb Z/n\mathbb Z) ⊕ (\mathbb Z/n\mathbb Z). Let \mathsf s_{≤ k}(G) be the smallest integer ℓ such that every sequence of ℓ terms from G, with repetition allowed, has a nonempty zero-sum subsequence with length at most k. It is known that \mathsf s_{≤ 2n-1-k}(G)=2n-1+k for k∈ [0,n-1], with the structure of extremal sequences showing this bound tight determined when k∈ \0,1,n-1\, and for various special cases when k∈ [2,n-2]. For the remaining values k∈ [2,n-2], the characterization of extremal sequences of length 2n-2+k avoiding a nonempty zero-sum of length at most 2n-1-k remained open in general, with it conjectured that they must all have the form e₁^[n-1] \boldsymbol⋅ e₂^[n-1] \boldsymbol⋅ (e₁ +e₂)^[k] for some basis (e₁,e₂) for G. Here x^[n] denotes a sequence consisting of the term x repeated n times. In this paper, we establish this conjecture for all k∈ [2,n-2] when n is prime, which in view of other recent work, implies the conjectured structure for all rank two abelian groups.

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• On zero-sum subsequences in a finite abelian group of length not exceeding a given number

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