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

Let \((G,+)\) be an abelian group and consider a subset \(A \subseteq G\) with \(|A|=k\). Given an ordering \((a_1, \ldots, a_k)\) of the elements of \(A\), define its {\em partial sums} by \(s_0 = 0\) and \(s_j = \sum_{i=1}^j a_i\) for \(1 \leq j \leq k\). We consider the following conjecture of Alspach: For any cyclic group \(\Z_n\) and any subset \(A \subseteq \Z_n \setminus \{0\}\) with \(s_k \neq 0\), 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 \(0 \leq i < j \leq k\). We show that Alspach's Conjecture holds for prime \(n\) when \(k \geq n-3\) and when \(k \leq 10\). 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 \(\Z_n \setminus \{0\}\) of size at least \(2k- \sqrt{8k}\) in all but at most a bounded number of cases.