{"title":"Rd的细子集中的点积树","authors":"A. Nadjimzadah","doi":"10.32523/2616-7182/bulmathenu.2022/2.2","DOIUrl":null,"url":null,"abstract":"A. Iosevich and K. Taylor showed that compact subsets of Rd with Hausdorff dimension greater than (d+1)/2 contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalentin a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of treeswith given gaps agrees with the regular value theorem.","PeriodicalId":286555,"journal":{"name":"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series","volume":"33 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Trees of Dot Products in Thin Subsets of Rd\",\"authors\":\"A. Nadjimzadah\",\"doi\":\"10.32523/2616-7182/bulmathenu.2022/2.2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A. Iosevich and K. Taylor showed that compact subsets of Rd with Hausdorff dimension greater than (d+1)/2 contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalentin a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of treeswith given gaps agrees with the regular value theorem.\",\"PeriodicalId\":286555,\"journal\":{\"name\":\"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series\",\"volume\":\"33 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-10-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.32523/2616-7182/bulmathenu.2022/2.2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.32523/2616-7182/bulmathenu.2022/2.2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
A. Iosevich和K. Taylor证明了Hausdorff维数大于(d+1)/2的Rd的紧子集包含开区间中有间隙的树。在相同的维数阈值下，我们证明了用点积代替距离的类似结果。此外，我们还证明了点积嵌入树的间隙在正Lebesgue测度集合中是普遍存在的，并且对于Ahlfors-David正则集，具有给定间隙的树的数量符合正则值定理。
A. Iosevich and K. Taylor showed that compact subsets of Rd with Hausdorff dimension greater than (d+1)/2 contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalentin a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of treeswith given gaps agrees with the regular value theorem.
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