Distinct partial sums in cyclic groups: polynomial method and constructive approaches

Let be an abelian group and consider a subset with . Given an ordering of the elements of , define its partial sums by and for . We consider the following conjecture of Alspach: for any cyclic group and any subset with , it is possible to find an ordering of the elements of such that no two of its partial sums and are equal for . We show that Alspach’s Conjecture holds for prime when and when . The former result is by direct construction, the latter is nonconstructive and uses the polynomial method. We also use the polynomial method to show that for prime a sequence of length having distinct partial sums exists in any subset of of size at least in all but at most a bounded number of cases.