Graham's rearrangement conjecture beyond the rectification barrier
Benjamin Bedert, Noah Kravitz
TL;DR
This work resolves Graham's rearrangement conjecture for a significantly larger class of subsets of $\mathbb{F}_p$ by proving that any $A$ with $|A|\le e^{c(\log p)^{1/4}}$ admits a two-sided valid ordering, improving the prior bound and revealing a quasi-polynomial threshold. The authors develop a structure theorem based on large dissociated subsets plus a rectifiable residual, enabling a rectification-and-randomization strategy that couples deterministic control with probabilistic anti-concentration across dissociated blocks. The proof proceeds through a four-step plan: (i) decompose $A$ into dissociated pieces and a small-dimension residual, (ii) order the positive and negative components to shield against zero-sum intervals, (iii) split and rearrange the dissociated blocks to create uniform- or anti-concentrated partial sums, and (iv) randomize inside each block to avoid end-interval zero-sums; this yields a high-probability construction of a two-sided valid ordering. The results extend to broad abelian groups with no nonzero elements of small order and point to several open problems, including improving the exponent, counting the number of valid orderings, and nonabelian generalizations.
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 $|A| \leqslant e^{(\log p)^{1/4}}$; 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.