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 an be cast in terms of the behavior of orbits of a suitable homogeneous flow. In particular, various phenomena related to the theory of integer equations or inequalities can be in a useful way interpreted as certain trajectories being generic with respect to some natural measures on homogeneous spaces, while other call for constructing trajectories with exceptional behavior. The principal investigator plans to continue his study of those orbits, aiming at new applications to number theory. Dynamical tools to be used are: measure rigidity, ergodic theorems, mixing and equidistribution, reduction theory, and quantitative nondivergence.
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 f sophisticated tools for their investigation.
" dealt with 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 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 project resulted in writing 11 original mathematical papers, out of which at the time of writing thsi report 10 are published and one is being refereed. Among the findings described in these papers are: - showing that several number theoretic and dynamical properties adhere to the so-called `almost all vs. no' dichotomy; - developing a theory of Modified Schmidt Games (MSGs) and proving that the set of weighted badly approximable vectors is a winning set of a certain MSG, deriving as a corollary results on countable intersections of those sets; - proving a conjecture made by Margulis in 1990 by modifying the winning set method of Schmidt, and generalizing it to proving that many dynamically defined sets are winning sets of MSGs; - proving a matrix version of Baker-Sprindzhuk conjectures in metric number theory (this generalizes work done by the PI and Margulis in late 1990s) - developing a technique of showing that certain dynamically and number-theoretically defined sets have large intersection with certain fractals by utilizing a theory of Schmidt games played on fractals. The PI also wrote an entry "Ergodic theory on homogeneous spaces and metric number theory" in Encyclopedia of Complexity and Systems Science, published by Springer, and a chapted in the Proceedings of the Clay Institute Summer School on Homogeneous Flows and Moduli Spaces. The PI has given lectures on topics related to the proect in many places around the world, including seminar talks at his home institution, which resulted in training students and generating interest to the field. Teaching several introductory graduate courses related to the subject of the grant contributed to education of graduate and undergraduate students at Brandeis. The PI was one of the organizers of the special session of the AMS national meeting in January 2009 (Washington, DC), a workshop on 'Homogeneous dynamics and number theory' at Oberwolfach in July 2010, and a session on 'Interactions between Dynamical Systems, Number Theory, and Combinatorics' at the AMS sectional meeting in Apil 2011, selecting many graduate students and postdocs to be invited to those events.
