Asymptotic and Uniform Diophantine Approximation Via Flows on Homogeneous Spaces

本课题研究某些代数动力系统及其在数论中的应用。在这里，动力系统代表着一组抽象的点，以及控制点随时间移动的方式的演化规律。这种抽象的动力系统构成了科学和工程中各种重要现象的模型的基础。结果表明，数学中的许多关于有理数同时逼近实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，这使得使用各种复杂的工具来研究它们成为可能。本研究项目旨在推进代数动力系统在逼近理论中的框架，发展新的方法，并对该领域的结果进行深远的推广。研究生和本科生将参与这个项目，并学习数论和动力学中的新方法和新技术。这些学生还将在素数计划的框架内指导研究项目，这是一个针对高中生的课后研究计划。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。这项研究项目继续研究动力学和数论中与渐近和一致近似有关的现象。将使用的数学工具包括：施密特博弈及其修改，Ekin-Marguis-Moze的积分不等式，有效混合和均匀分布性质，Siegel-Rogers矩公式，以及数的参数几何。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.