An improved result for Falconer’s distance set problem in even dimensions

By Xiumin Du, Alex Iosevich, Yumeng Ou, Hong Wang, Ruixiang Zhang. Mathematische Annalen, 2020

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We show that if compact set E⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subset \mathbb {R}^d$$\end{document} has Hausdorff dimension larger than d2+14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d}{2}+\frac{1}{4}$$\end{document}, where d≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 4$$\end{document} is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer’s distance set conjecture in even dimensions.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ce2a37450ace567e5c11cbd912b3d0d59e1b2577