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Asymptotic vs. Uniform Approximation in Dynamical Systems and Number Theory

动力系统和数论中的渐近与一致逼近

基本信息

批准号：
1900560
负责人：
Dmitry Kleinbock
金额：
$ 22.3万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900560&HistoricalAwards=false
关键词：
Asymptotic vs.Uniform Approximation Dynamical

项目摘要

This project deals with certain algebraic dynamical systems 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 also many problems 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 has been an influx of new ideas concerning connections between Diophantine approximation and dynamical systems. This research project continues the study of phenomena in both homogeneous dynamics and number theory in two directions: related to so-called asymptotic and uniform approximation. Among mathematical tools to be employed are: quantitative non-divergence on the space of lattices, Schmidt games and their modifications, integral inequalities of Eskin-Margulis-Mozes, exponential mixing, equidistribution properties, and 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.

这个项目涉及某些代数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，数学中的许多问题也可以用动力系统的行为来理解，这些问题涉及到用有理数同时逼近真实的数。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，出现了大量的新思想。这个研究项目继续在两个方向研究齐次动力学和数论中的现象：与所谓的渐近和一致逼近有关。数学工具包括：格空间上的定量不发散、施密特博弈及其修正、Eskin-Margulis-Mozes积分不等式、指数混合、等分布性质和数字几何。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标准进行评估，被认为值得支持。