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.
