A new proof of Chen's theorem for Markoff graphs
成果类型：
Article
署名作者：
Martin, Daniel E.
署名单位：
Clemson University
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910
DOI：
10.1007/s00222-025-01346-9
发表日期：
2025
页码：
623-626
关键词：
摘要：
In 2021, Chen proved that the size of any connected component of the Markoff mod p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document} graph is divisible by p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen's result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document} graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen's theorem.