{"title":"格雷厄姆重排猜想超越整流障碍","authors":"Benjamin Bedert, Noah Kravitz","doi":"arxiv-2409.07403","DOIUrl":null,"url":null,"abstract":"A 1971 conjecture of Graham (later repeated by Erd\\H{o}s and Graham) asserts\nthat every set $A \\subseteq \\mathbb{F}_p \\setminus \\{0\\}$ has an ordering whose\npartial sums are all distinct. We prove this conjecture for sets of size $|A|\n\\leqslant e^{(\\log p)^{1/4}}$; our result improves the previous bound of $\\log\np/\\log \\log p$. One ingredient in our argument is a structure theorem involving\ndissociated sets, which may be of independent interest.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"110 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Graham's rearrangement conjecture beyond the rectification barrier\",\"authors\":\"Benjamin Bedert, Noah Kravitz\",\"doi\":\"arxiv-2409.07403\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A 1971 conjecture of Graham (later repeated by Erd\\\\H{o}s and Graham) asserts\\nthat every set $A \\\\subseteq \\\\mathbb{F}_p \\\\setminus \\\\{0\\\\}$ has an ordering whose\\npartial sums are all distinct. We prove this conjecture for sets of size $|A|\\n\\\\leqslant e^{(\\\\log p)^{1/4}}$; our result improves the previous bound of $\\\\log\\np/\\\\log \\\\log p$. One ingredient in our argument is a structure theorem involving\\ndissociated sets, which may be of independent interest.\",\"PeriodicalId\":501064,\"journal\":{\"name\":\"arXiv - MATH - Number Theory\",\"volume\":\"110 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Number Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.07403\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07403","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.