{"title":"仿射子空间并集和Furstenberg型集的Hausdorff维数","authors":"K. Héra, Tamás Keleti, András Máthé","doi":"10.4171/JFG/77","DOIUrl":null,"url":null,"abstract":"We prove that for any $1 \\le k<n$ and $s\\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < \\alpha \\le k$, if $B \\subset {\\mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${\\mathbb R}^n$ such that every $P \\in E$ intersects $B$ in a set of Hausdorff dimension at least $\\alpha$, then $\\dim B \\ge 2 \\alpha - k + \\min(\\dim E, 1)$, where $\\dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well known Furstenberg-type estimate that every $\\alpha$-Furstenberg set has Hausdorff dimension at least $2 \\alpha$; we strengthen a theorem of Falconer and Mattila; and we show that for any $0 \\le k<n$, if a set $A \\subset {\\mathbb R}^n$ contains the $k$-skeleton of a rotated unit cube around every point of ${\\mathbb R}^n$, or if $A$ contains a $k$-dimensional affine subspace at a fixed positive distance from every point of ${\\mathbb R}^n$, then the Hausdorff dimension of $A$ is at least $k + 1$.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2017-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/77","citationCount":"24","resultStr":"{\"title\":\"Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets\",\"authors\":\"K. Héra, Tamás Keleti, András Máthé\",\"doi\":\"10.4171/JFG/77\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that for any $1 \\\\le k<n$ and $s\\\\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\\\\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < \\\\alpha \\\\le k$, if $B \\\\subset {\\\\mathbb R}^n$ and $E$ is a nonempty collection of $k$-dimensional affine subspaces of ${\\\\mathbb R}^n$ such that every $P \\\\in E$ intersects $B$ in a set of Hausdorff dimension at least $\\\\alpha$, then $\\\\dim B \\\\ge 2 \\\\alpha - k + \\\\min(\\\\dim E, 1)$, where $\\\\dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well known Furstenberg-type estimate that every $\\\\alpha$-Furstenberg set has Hausdorff dimension at least $2 \\\\alpha$; we strengthen a theorem of Falconer and Mattila; and we show that for any $0 \\\\le k<n$, if a set $A \\\\subset {\\\\mathbb R}^n$ contains the $k$-skeleton of a rotated unit cube around every point of ${\\\\mathbb R}^n$, or if $A$ contains a $k$-dimensional affine subspace at a fixed positive distance from every point of ${\\\\mathbb R}^n$, then the Hausdorff dimension of $A$ is at least $k + 1$.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2017-01-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.4171/JFG/77\",\"citationCount\":\"24\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/JFG/77\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/JFG/77","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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