On Falconer's distance set problem in the plane

• 成果类型: Article
• 署名作者: Guth, Larry; Iosevich, Alex; Ou, Yumeng; Wang, Hong
• 署名单位: Massachusetts Institute of Technology (MIT); University of Rochester; City University of New York (CUNY) System; Baruch College (CUNY)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00917-x
• 发表日期: 2020
• 页码: 779-830
• 关键词: hausdorff dimension; fourier-transforms; averages

摘要

If E subset of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}<^>2$$\end{document} is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point x is an element of E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in E$$\end{document} so that the set of distances {|x-y|}y is an element of E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ |x-y| \}_{y \in E}$$\end{document} has positive Lebesgue measure.