On Graham's rearrangement conjecture

Abstract

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq {\mathbb{Z}}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots ,s_{|S|}$ where all partial sums $s_1,s_1+s_2,\dots ,s_1+s_2+\cdots +s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq {\mathbb{Z}}_p\setminus \{0\}$ with $|S|\le p^{1-\hat{I}\pm }$ and $|S|$ sufficiently large with respect to $\hat{I}\pm$, for any $\hat{I}\pm \in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$.