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Graham's rearrangement conjecture beyond the rectification barrier (2409.07403v2)
Published 11 Sep 2024 in math.CO and math.NT
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.