Structure of a sequence with prescribed zero-sum subsequences: Rank two [formula omitted]-groups

Abstract

Let G=(Z/nZ)⊕(Z/nZ). Let 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 s≤2n−1−k(G)=2n−1+k for k∈[0,n−1]. The structure of extremal sequences that show this bound is tight was determined for 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. It is conjectured that they must all have the form e1[n−1]⋅e2[n−1]⋅(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 abelian groups of rank two.