Graham's rearrangement conjecture beyond the rectification barrier

TL;DR

This paper proves Graham's rearrangement conjecture for certain large sets in finite fields, extending previous bounds and introducing a new structure theorem involving dissociated sets.

Contribution

It advances the understanding of Graham's conjecture by establishing it for larger sets and presents a novel structure theorem related to dissociated sets.

Findings

Proves Graham's conjecture for sets up to size e^{(log p)^{1/4}}

Improves previous bounds from log p / log log p

Introduces a structure theorem involving dissociated sets

Abstract

A 1971 conjecture of Graham (later repeated by Erd\H{o}s 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 $|A| \leqslant e^{(\log p)^{1/4}}$; 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.