On Falconer's distance set problem in the plane

摘要

If E subset of R2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{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 Edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$x in E$$end{document} so that the set of distances {|x-y|}y is an element of Edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${ |x-y| }_{y in E}$$end{document} has positive Lebesgue measure.