Course Objectives: In this course we discuss some of the results and conjectures in number theory that can be approached using techniques from the theory of dynamical systems. For instance, we consider the following questions:

• Given a vector with real coordinates, how well can it be approximated by vectors with rational coordinates?
• Is the sequence of real numbers (3/2)ⁿ dense modulo one? What about the double sequence 2ⁿ3^mx modulo one for irrational x?
• Given a polynomial function, is the set of its values at integral points dense/discrete in the real lines?
• What is the behaviour of the high-energy eigenstates of the Laplace operator?

It turns out that all these deep questions are intimately related to distribution of orbits of suitable dynamical systems. While this is also a difficult problem on its own right, there has been several recent breakthroughs in the field that we shall discuss. In particular, one of the highlights of the course will be Ratner's theorems about the properties of unipotent flows and their applications to number theory.

The subject of the course lies on crossroads between the theory of dynamical systems and number theory. However, we assume only minimal background in these fields and rely mostly on basic knowledge of analysis and algebra.

We plan to cover some of the following topics:

• Flows on the space of lattices, geodesic and horocycle flows and their properties,
• Howe-Moore theorem on decay of matrix coefficients,
• Elements of the theory of Diophantine approximation and its connections with dynamical systems,
• Ratner's theorem on unipotent flows,
• Oppenheim conjecture on distribution of values of quadratic forms,
• Baker-Sprindzhuk conjecture on Diophantine approximation on manifolds,
• Actions of higher-rank abelean groups and Furstenberg's conjecture,
• Littlewood and Schmidt conjectures on simultaneous Diophantine approximation.
• Quantum ergodicity and quantum unique ergodicity on arithmetic surfaces.

Time: Monday 2-4pm; See TCC Timetable

Homework Problems:

• Homework 1
• Homework 2
• Homework 3

Course Assessment: The course will be assessed by problem sheets that will be posted here.

References:

• B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces. London Mathematical Society Lecture Note Series, 269. Cambridge University Press, Cambridge, 2000.
• M. Einsiedler, Ratner's theorem for SL(2,R)-invariant measures.
• M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces.
• M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Springer Graduate Text in Mathematics, 2010.
• M. Einsiedler and T. Ward, Arithmetic quantum unique ergodicity.
• D. Kleinbock, Some applications of homogeneous dynamics to number theory. Smooth ergodic theory and its applications (Seattle, WA, 1999), 639--660, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.
• D. Kleinbock, Metric Diophantine approximation and dynamical systems.
• E. Lindenstrauss, Some examples how to use measure classification in number theory. Equidistribution in number theory (A. Granville and Z. Rudnick, eds.), 261-303, Springer, 2007.
• G. Margulis, Diophantine approximation, lattices and flows on homogeneous spaces. A panorama of number theory or the view from Baker's garden (Zurich, 1999), 280-310, Cambridge Univ. Press, Cambridge, 2002.
• A. Starkov, Dynamical systems on homogeneous spaces. Translations of Mathematical Monographs, 190. American Mathematical Society, Providence, RI, 2000.
• A. Venkatesh, The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture. Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 117-134.
• D. Witte Morris, Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005.