Dynamical Approaches to Number Theory and Additive Combinatorics

点击翻译按钮获取中文摘要

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.