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

TL;DR

This paper investigates Alspach's conjecture on ordering elements in cyclic groups to produce distinct partial sums, proving it for large and small subset sizes using constructive and polynomial methods.

Contribution

It proves Alspach's conjecture for prime cyclic groups when subset size is large or small, introducing new constructive and polynomial techniques.

Findings

Confirmed the conjecture for prime $n$ with $k geq n-3$ and $k leq 10$

Developed a polynomial method to find sequences with distinct partial sums in large subsets

Established bounds on subset sizes guaranteeing the existence of sequences with distinct partial sums

Abstract

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.