Distinct Partial Sums in Cyclic Groups: Polynomial Method and Constructive Approaches

Abstract

Let (G, +) be an abelian group and consider a subset A ⊆ G with |A| = k. Given an ordering (a 1 , . . . , a k ) of the elements of A, define its partial sums by s 0 = 0 and s j = j i=1 a i for 1 ≤ j ≤ k. We consider the following conjecture of Alspach: For any cyclic group Z n and any subset A ⊆ Z n \ {0} with s k = 0, it is possible to find an ordering of the elements of A such that no two of its partial sums s i and s j are equal for 0 ≤ i < j ≤ k. We show that Alspach's Conjecture holds for prime n when k ≥ n − 3 and when k ≤ 10. The former result is by direct construction, the latter is non-constructive and uses the polynomial method. We also use the polynomial method to show that for prime n a sequence of length k having distinct partial sums exists in any subset of Z n \ {0} of size at least 2k − √ 8k in all but at most a bounded number of cases.