Old and new techniques in homogeneous dynamics and Diophantine approximation: quantitative nondivergence, Schmidt games, random walks

齐次动力学和丢番图近似中的新旧技术：定量非散度、施密特游戏、随机游走

• 批准号: 1101320
• 负责人: Dmitry Kleinbock
• 金额: $ 18.21万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101320&HistoricalAwards=false
• 关键词: Old; new; techniques; homogeneous; dynamics

This project deals with algebraic dynamical systems and their applications to number theory. It has been observed for a long time that many problems concerning diophantine approximation, that is, various phenomena related to the theory of integer equations or inequalities, can be cast in terms of the behavior of trajectories of a suitable homogeneous flow. The principal investigator plans to continue his study of those trajectories, aiming at new applications to number theory. Dynamical tools to be used are: quantitative nondivergence on the space of lattices, Schmidt games and their modifications, recurrence of random walks, measure rigidity, mixing and equidistribution properties. A dynamical system here stands for an abstract set of points together with an evolution law which governs the way points move over time. It turns out that many problems concerning simultaneous approximation of real numbers by rational numbers can be understood in terms of the behavior of certain orbits. 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. The educational aspect of the proposal involves exposing graduate and undergraduate students to new methods and techniques in number theory and dynamical systems, and supervising research projects of high school students within the framework of the PRIMES program.

这个项目涉及代数动力系统及其在数论中的应用。长期以来，人们已经注意到，许多关于丢番图逼近的问题，即与整数方程或不等式理论有关的各种现象，都可以用适当的均匀流的轨迹的行为来描述。首席研究员计划继续研究这些轨迹，旨在数论的新应用。要使用的动态工具是：定量nondivergence空间的格子，施密特游戏及其修改，复发的随机游走，措施刚性，混合和equidistribution属性。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。事实证明，许多关于用有理数同时逼近真实的数的问题，可以用某些轨道的行为来理解。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。该提案的教育方面涉及让研究生和本科生接触数论和动力系统中的新方法和技术，并在PRIMES计划框架内监督高中生的研究项目。