Representation of group elements as subsequence sums

Abstract

Let G be a finite (additive written) abelian group of order n . Let w 1 , … , w n be integers coprime to n such that w 1 + w 2 + ⋯ + w n ≡ 0 (mod n). Let I be a set of cardinality 2 n - 1 and let ξ = { x i : i ∈ I } be a sequence of elements of G. Suppose that for every subgroup H of G and every a ∈ G , ξ contains at most 2 n - n | H | terms in a + H . Then, for every y ∈ G , there is a subsequence { y 1 , … , y n } of ξ such that y = w 1 y 1 + ⋯ + w n y n . Our result implies some known generalizations of the Erdős–Ginzburg–Ziv Theorem.