Graham's rearrangement conjecture beyond the rectification barrier

Abstract

A 1971 conjecture of Graham (later repeated by Erdős and Graham) asserts that every set $A\subseteq {\mathbb{F}}_p\setminus \{0\}$ has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size $|A|⩽e^{(\log p{)}^{1/4}}$; our result improves the previous bound of $\log p/\log \log p$. One ingredient in our argument is a structure theorem involving dissociated sets, which may be of independent interest.

Incorporates referee's suggestions