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

本课题研究代数动力系统及其在数论中的应用。长期以来，人们观察到许多与丢番图近似有关的问题，即与整数方程或不等式理论有关的各种现象，都可以根据适当的均质流动的轨迹行为来描述。首席研究员计划继续他对这些轨迹的研究，目标是数论的新应用。可使用的动力学工具包括：格空间上的定量无散度、Schmidt对策及其修改、随机游动的常返性、测度刚性、混合和均匀分布性质。在这里，动力系统代表一组抽象的点，以及支配点随时间移动的方式的演化规律。结果表明，许多关于有理数同时逼近实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，这使得使用各种复杂的工具来研究它们成为可能。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的新方法和新技术，并在素数计划的框架内监督高中生的研究项目。

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.