Zero-sum subsequences of length kqkq over finite abelian pp-groups

For a finite abelian group GG and a positive integer kk, let sk(G) denote the smallest integer ℓ∈Nℓ∈N such that any sequence SS of elements of GG of length |S|≥ℓ|S|≥ℓ has a zero-sum subsequence with length kk. The celebrated Erdős–Ginzburg–Ziv theorem determines sn(Cn)=2n−1 for cyclic groups CnCn, while Reiher showed in 2007 that sn(Cn2)=4n−3. In this paper we prove for a pp-group GG with exponent exp(G)=qexp(G)=q the upper bound skq(G)≤(k+2d−2)q+3D(G)−3 whenever k≥dk≥d, where d=⌈D(G)q⌉ and pp is a prime satisfying p≥2d+3⌈D(G)2q⌉−3, where D(G) is the Davenport constant of the finite abelian group GG. This is the correct order of growth in both kk and dd. Subject to the same assumptions, we show exact equality skq(G)=kq+D(G)−1 if k≥p+dk≥p+d and p≥4d−2p≥4d−2, resolving a case of the conjecture of Gao, Han, Peng, and Sun that skexp(G)(G)=kexp(G)+D(G)−1 whenever kexp(G)≥D(G). We also obtain a general bound skn(Cnd)≤9kn for nn with large prime factors and kk sufficiently large. Our methods extend the algebraic method of Kubertin, who proved that skq(Cqd)≤(k+Cd2)q−d if k≥dk≥d and qq is a prime power.