Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

这一建议包含许多不同的问题，其共同主线是结合了动力学、概率论和主要在经典遍历理论环境中开发的一组基本技术的核心方法。这些技术已经非常成功地适用于困难分析和组合学中的问题，这些领域包括：KAM理论；Ramsey理论(特别是以前已知的关于划分的结果的密度版本，例如，Hales-Julett定理的密度版本)；欧几里德空间中集合的分形几何和Haussdorff维度；从属群及其作用的理论。这些是我们建议进一步发展的主要领域。满足同一组定律的物理系统表现出各种各样的行为，从看起来完全随机的完全混沌系统的极端到表现出令人惊讶的稳定水平的系统(例如，粒子在回旋加速器中运动)。数学动力学和遍历理论解释了这些不同类型的行为。同样的方法在其他领域也被证明是非常有用的，特别是在组合学、信息论、数据压缩等方面。我们的工作正在开发新的方法，以更好地解释这些现象，并将该领域的适用性扩展到更广泛的问题类别。

This proposal contains a number of different problems whose common thread is a core approach combining dynamics, probability theory, and a basic set of techniques that were mainly developed in the classical ergodic theoretic setting. These techniques have been very successfully adapted to problems from hard analysis and combinatorics, in areas such as: KAM theory; Ramsey theory (in particular density versions of results previously known for partitions, e.g., the density version of the Hales-Jewett theorem); Fractal geometry and Haussdorff dimension of sets in Euclidean space; the theory of amenable groups and their actions. These are the main areas that we propose to develop further.Physical systems satisfying the same set of laws exhibit a wide variety of behaviors going from the extreme of completely chaotic systems which appear to be completely random to systems that exhibit a surprising level of stability (for instance, a particle moving in a cyclotron). Mathematical dynamics and ergodic theory explain these different types of behaviors. The same methods turn out to be extremely useful in other areas, in particular in combinatorics, information theory, data compression, etc. Our work is developing new methods that give better explanations of these phenomena and extend the applicability of the field to a broader class of problems.