Structure of a sequence with prescribed zero-sum subsequences: Rank Two $p$-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 thatwith 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 e [n−1] 1 • e [n−1] 2• (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.