Representation of group elements as subsequence sums

Abstract

AbstractLet G be a finite (additive written) abelian group of order n. Let w1,…,wn be integers coprime to n such that w1+w2+⋯+wn≡0 (mod n). Let I be a set of cardinality 2n-1 and let ξ={xi: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 2n-n|H| terms in a+H.Then, for every y∈G, there is a subsequence {y1,…,yn} of ξ such that y=w1y1+⋯+wnyn.Our result implies some known generalizations of the Erdős–Ginzburg–Ziv Theorem