{"title":"逆闵可夫斯基定理","authors":"O. Regev, Noah Stephens-Davidowitz","doi":"10.1145/3055399.3055434","DOIUrl":null,"url":null,"abstract":"We prove a conjecture due to Dadush, showing that if ℒ⊂ ℝn is a lattice such that det(ℒ′) 1 for all sublattices ℒ′ ⊆ ℒ, then $$\\sum_{y∈ℒ}^e-t2||y||2≤3/2,$$ where t := 10(logn + 2). From this we also derive bounds on the number of short lattice vectors and on the covering radius.","PeriodicalId":20615,"journal":{"name":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","volume":"32 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2016-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"28","resultStr":"{\"title\":\"A reverse Minkowski theorem\",\"authors\":\"O. Regev, Noah Stephens-Davidowitz\",\"doi\":\"10.1145/3055399.3055434\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove a conjecture due to Dadush, showing that if ℒ⊂ ℝn is a lattice such that det(ℒ′) 1 for all sublattices ℒ′ ⊆ ℒ, then $$\\\\sum_{y∈ℒ}^e-t2||y||2≤3/2,$$ where t := 10(logn + 2). From this we also derive bounds on the number of short lattice vectors and on the covering radius.\",\"PeriodicalId\":20615,\"journal\":{\"name\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-11-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"28\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3055399.3055434\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3055399.3055434","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove a conjecture due to Dadush, showing that if ℒ⊂ ℝn is a lattice such that det(ℒ′) 1 for all sublattices ℒ′ ⊆ ℒ, then $$\sum_{y∈ℒ}^e-t2||y||2≤3/2,$$ where t := 10(logn + 2). From this we also derive bounds on the number of short lattice vectors and on the covering radius.
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