Asymptotic and Uniform Diophantine Approximation Via Flows on Homogeneous Spaces

这个项目涉及某些代数起源的动力系统及其在数论中的应用。这里的动力系统代表一组抽象的点，以及一个控制点随时间移动的演化定律。这种抽象的动力系统形成了科学和工程中广泛的重要现象模型的基础。事实证明，数学中许多关于用有理数同时逼近真实的数的问题都可以用动力系统的行为来理解。此外，在这种情况下出现的系统是代数性质的，这使得有可能使用各种各样的复杂工具进行调查。该研究项目旨在推进近似理论中代数动力系统的框架，开发新方法，并在该领域获得具有深远意义的推广结果。研究生和本科生将参与该项目，并介绍数论和动力学的新方法和技术。这些学生还将在PRIMES计划的框架内监督研究项目，PRIMES计划是一项针对高中生的课外研究计划。近年来，关于丢番图近似和动力系统之间的联系有了重要的发展。这个研究项目继续研究与渐近和一致近似有关的动力学和数论现象。其中的数学工具将被采用：施密特游戏及其修改，积分不等式的埃斯金-马古利斯-莫泽斯，有效的混合和均匀分布的性质，西格尔-罗杰斯矩公式，参数几何的数字。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This project deals with certain dynamical systems of algebraic origin and their applications to number theory. A dynamical system here stands for an abstract set of points together with an evolution law that governs the way points move over time. Such abstract dynamical systems form the basis for models of a wide range of important phenomena in science and engineering. It turns out that many questions in mathematics concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of dynamical systems. Furthermore, systems that arise in this context are of algebraic nature, which makes it possible to use a wide variety of sophisticated tools for their investigation. This research project aims to advance the framework of algebraic dynamical systems in approximation theory, develop new methods, and obtain far-reaching generalizations of results in the field. Graduate and undergraduate students will be involved in the project and introduced to new methods and techniques in number theory and dynamics. These students will also supervise research projects within the framework of the PRIMES program, an after-school research program for high school students.During recent years there have been important developments concerning connections between Diophantine approximation and dynamical systems. This research project continues the study of phenomena in both dynamics and number theory related to asymptotic and uniform approximations. Among the mathematical tools to be employed are: Schmidt games and their modifications, integral inequalities of Eskin-Margulis-Mozes, effective mixing and equidistribution properties, Siegel-Rogers moment formulas, and parametric geometry of numbers.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.