Rearranging small sets for distinct partial sums

TL;DR

The paper proves Graham's conjecture for small sets in finite fields, showing such sets can be ordered to have all distinct partial sums, using a concise proof for sets up to size logarithmic in p.

Contribution

Provides a short proof confirming Graham's conjecture for sets of size up to logarithmic in p, advancing understanding of partial sum arrangements in finite fields.

Findings

Confirmed Graham's conjecture for small sets in finite fields.

Developed a concise proof applicable to sets of size up to log p / log log p.

Enhanced theoretical understanding of ordering elements for distinct partial sums.

Abstract

A conjecture of Graham (repeated by Erd\H{o}s) asserts that for any set $A \subseteq \mathbb{F}_p \setminus \{0\}$, there is an ordering $a_1, \ldots, a_{|A|}$ of the elements of $A$ such that the partial sums $a_1, a_1+a_2, \ldots, a_1+a_2+\cdots+a_{|A|}$ are all distinct. We give a very short proof of this conjecture for sets $A$ of size at most $\log p/\log\log p$.