Dynamical Approaches to Number Theory and Additive Combinatorics

在过去的几十年里，用动态方法解决拉姆齐理论、加性组合学和数论中出现的问题已经相当成功。由PI和其他研究人员开创的一些最新技术，扩大了这些思想的应用范围，解决了一些以前无法解决的问题。本建议旨在以这些新颖的想法为基础，以便对这些领域的一些基本问题获得新的见解。我们拟研究的问题范围包括：多项式组形在自然数中的配分正则性问题；关于乘法函数的统计行为的问题，以及在每一组正密度中存在无穷构型的问题。尽管看起来不相关，但所有这些问题之间都有很深的联系，通常用遍历理论或动力系统的语言来表述。

In the last few decades, dynamical approaches to problems arising in Ramsey theory, Additive Combinatorics and Number theory have been rather successful. Some very recent techniques, pioneered by the PI and other researchers, widen the range of applications of these ideas to some problems previously out of reach. This proposal seeks to build upon these novel ideas in order to gain new insight into some fundamental problems in those areas.The range of problems we propose to investigate include questions about partition regularity of polynomial configurations in the natural numbers; questions about the statistical behavior of multiplicative functions, and the question of which infinite configurations are present in every set of positive density.Despite looking unrelated, there are deep connections between all these problems, often formulated in the language of ergodic theory or dynamical systems.