Zero-sum subsequences of length [formula omitted] over finite abelian [formula omitted]-groups

Abstract

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