{"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":48484,"journal":{"name":"Journal of Fractal Geometry","volume":" ","pages":""},"PeriodicalIF":1.1000,"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\":48484,\"journal\":{\"name\":\"Journal of Fractal Geometry\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"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\":\"Journal of Fractal Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4171/JFG/77\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Fractal Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/JFG/77","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}