Graham’s rearrangement conjecture beyond the rectification barrier

A 1971 conjecture of Graham (later repeated by Erdős and Graham) asserts that every set A ⊆ Fₚ \; \; \0\ A ⊆ F p ∖ { 0 } has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size |A| e^(log p)1/4 ∣ 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.