格雷厄姆重排猜想超越整流障碍

Benjamin Bedert, Noah Kravitz
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引用次数: 0

摘要

格雷厄姆在 1971 年提出的一个猜想(后来被埃尔德和格雷厄姆重复)断言,每个集合 $A \subseteq \mathbb{F}_p \setminus \{0\}$都有一个排序,其部分和都是不同的。我们针对大小为 $|A|leqslant e^{(\log p)^{1/4}}$ 的集合证明了这一猜想;我们的结果改进了之前的$\logp/\log \log p$ 的约束。我们论证的一个要素是一个涉及关联集的结构定理,这可能是我们感兴趣的。
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Graham's rearrangement conjecture beyond the rectification barrier
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.
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