A Multidimensional Szemerédi Theorem in the Primes via Combinatorics
A Multidimensional Szemerédi Theorem in the Primes via Combinatorics
复制标题
基于组合学的素数多维 Szemeredi 定理
DOI：
--
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发表时间：
2013
影响因子：
0.5
通讯作者：
Tatchai Titichetrakun
中科院分区：
文献类型：
--
作者：
Brian Cook;Á. Magyar;Tatchai Titichetrakun
Let A be a subset of positive relative upper density of Pd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^d$$\end{document}, the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set F⊆Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\subseteq \mathbb {Z}^d$$\end{document}. This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.
Let A be a subset of positive relative upper density of Pd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^d$$\end{document}, the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set F⊆Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\subseteq \mathbb {Z}^d$$\end{document}. This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.