The Number of k-Sums Modulo k Original Research Article

Let a1, …, ar be a sequence of elements of imagek, the integers modulo k. Calling the sum of k terms of the sequence a k-sum, how small can the set of k-sums be? Our aim in this paper is to show that if 0 is not a k-sum then there are at least r−k+1 k-sums. This result, which is best possible, extends the Erdimages–Ginzburg–Ziv theorem, which states that if r=2k−1 then 0 is a k-sum. We also show that the same result holds in any abelian group of order k, and make some related conjectures.