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Asymptotic and Uniform Diophantine Approximation Via Flows on Homogeneous Spaces

通过齐次空间上的流进行渐近一致丢番图逼近

基本信息

批准号：
2155111
负责人：
Dmitry Kleinbock
金额：
$ 39.55万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2155111&HistoricalAwards=false
关键词：
Asymptotic Uniform Diophantine Approximation Via

项目摘要

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.

该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控制点随时间移动方式的演化定律。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。事实证明，数学中有关有理数同时逼近实数的许多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具有代数性质，这使得使用各种复杂的工具进行研究成为可能。该研究项目旨在推进近似理论中的代数动力系统框架，开发新方法，并获得该领域成果的深远推广。研究生和本科生将参与该项目，并了解数论和动力学方面的新方法和技术。这些学生还将监督 PRIMES 计划框架内的研究项目，这是一项针对高中生的课后研究计划。近年来，丢番图近似与动力系统之间的联系取得了重要进展。该研究项目继续研究与渐近和一致近似相关的动力学和数论现象。所采用的数学工具包括：施密特博弈及其修改、Eskin-Margulis-Mozes 积分不等式、有效混合和均匀分布特性、Siegel-Rogers 矩公式和数字参数几何。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。