On weighted sums in abelian groups Original Research Article

Let G be an abelian group of order n and Davenport constant d and let k be a natural number. Let x0,x1, …,xm be a sequence of elements of G such that xo has the most repeated value in the sequence. Let {wi; 1 ⩽ i ⩽ k} be a family of integers prime relative to n. We obtain the following two generalizations of the Erdös-Ginzburg-Ziv Theorem. For m ⩾ n + k − 1, we prove that there is a permutation α of [1,m] such that ∑1 ⩽ i ⩽ kwixα(i)=∑1 ⩽ i ⩽ kwix0. For k ⩾ n − 1 and m ⩾ k + d − 1, we prove that there is a k-subset K ⊂ [1, m] such that ∑i ∈ Kxi=kx0.