Dynamical Approaches to Number Theory and Additive Combinatorics

在过去的几十年里，对Ramsey理论、加性组合学和数论中出现的问题的动力学方法都是相当成功的。由PI和其他研究人员开创的一些非常新的技术，扩大了这些想法在一些以前无法解决的问题上的应用范围。这一建议试图建立在这些新思想的基础上，以期对这些领域的一些基本问题有新的认识。我们建议研究的问题包括自然数中多项式构形的划分正则性问题，乘性函数的统计行为问题，以及在每一组正密度中存在哪些无限构形的问题。尽管看起来不相关，但所有这些问题之间都有深刻的联系，通常是用遍历理论或动力系统的语言来表述的。

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.