Investigations in Combinatorics and Number Theory via Ergodic Theoretic Methods

历史上，动力系统理论植根于对宏观物体（如卫星、行星等）和微观粒子（如分子和原子）运动的研究。今天，这是一个广阔的领域，追求轨道和一般系统随时间演变的变化的研究。它是一个丰富而活跃的研究领域，不仅在当代数学中发挥着重要作用，而且对其他科学也有很大的贡献，并有许多应用。该项目旨在改进动力系统中可用的分析工具和技术，预计将在广泛的领域产生影响。这个项目的主要目标集中在纯数学中三个不同领域的接口：动力系统理论、组合学和数论。从可测量的、拓扑的和符号的动态中衍生出来的分析工具的使用通常提供了分析数论和组合情况的新方法。事实证明，这种方法在解决看似棘手的问题方面非常有效。同样的，这个项目的目的是在动态方法的帮助下，解决和潜在地解决算术组合学和数论中的猜想和开放问题。该项目的主要挑战范围从用动态语言重铸离散数学中的问题到开发相关的动态工具来解决这些重新表述。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Historically, the theory of dynamical systems was rooted in the study of the movement of macroscopic objects (such as moons, planets, etc.) and microscopic particles (such as molecules and atoms). Today, it is a vast area that pursues the study of orbits and transformations in general systems evolving with time. It is a rich and active research area that not only plays an important role in contemporary mathematics, but also greatly contributes to other sciences and finds many applications. This project seeks to improve upon the analytic tools and techniques available in dynamical systems, with anticipated impact in a wide range of fields.The primary objectives of this project are concentrated at the interface of three different areas in pure mathematics: the theory of dynamical systems, combinatorics, and number theory. The employment of analytic tools stemming from measurable, topological, and symbolic dynamics often offers novel ways to analyze number-theoretic and combinatorial situations. This approach has proven to be powerful in solving seemingly intractable problems. In the same vein, this project aims to address, and potentially settle, conjectures and open questions in arithmetic combinatorics and number theory with the help of dynamical methods. The main challenges of the project range from recasting problems in discrete mathematics in a dynamical language to developing the pertinent dynamical tools for solving these reformulations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.