Unions of lines in R n $\mathbb {R}^n$

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematika Pub Date : 2023-02-17 DOI:10.1112/mtk.12190

Joshua Zahl

{"title":"Unions of lines in \n \n \n R\n n\n \n $\\mathbb {R}^n$","authors":"Joshua Zahl","doi":"10.1112/mtk.12190","DOIUrl":null,"url":null,"abstract":"We prove a conjecture of D. Oberlin on the dimension of unions of lines in <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math>. If <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$d\\geqslant 1$</annotation>\n </semantics></math> is an integer, <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo>⩽</mo>\n <mi>β</mi>\n <mo>⩽</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$0\\leqslant \\beta \\leqslant 1$</annotation>\n </semantics></math>, and L is a set of lines in <math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math> with Hausdorff dimension at least <math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mo>(</mo>\n <mi>d</mi>\n <mo>−</mo>\n <mn>1</mn>\n <mo>)</mo>\n <mo>+</mo>\n <mi>β</mi>\n </mrow>\n <annotation>$2(d-1)+\\beta$</annotation>\n </semantics></math>, then the union of the lines in L has Hausdorff dimension at least <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>+</mo>\n <mi>β</mi>\n </mrow>\n <annotation>$d + \\beta$</annotation>\n </semantics></math>. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"473-481"},"PeriodicalIF":0.8000,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12190","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12190","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove a conjecture of D. Oberlin on the dimension of unions of lines in $R^{n}$ . If $d ⩾ 1$ is an integer, $0 ⩽ β ⩽ 1$ , and L is a set of lines in $R^{n}$ with Hausdorff dimension at least $2 (d - 1) + β$ , then the union of the lines in L has Hausdorff dimension at least $d + β$ . Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.

查看原文本刊更多论文

Rn$\mathbb {R}^n$中的行并集

我们证明了D.Oberlin关于Rn$\mathbb｛R｝^n$中直线并集维数的一个猜想。如果d⩾1$d\geqslant 1$是一个整数，0⩽β\108777 1$0\leqslant\beta\leqslant 1$，并且L是Rn$\mathbb｛R｝^n$中的一组线，其中Hausdorff维数至少为2（d−1）+β$2（d-1）+\beta$，则L中的线的并集具有至少d+β$d+\beta$Hausdorf维数。我们的证明结合了Carbery和Valdimarsson对多重线性Kakeya定理的改进版本和多重线性→ Bourgain和Guth的线性论证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematika MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.