The main goal of this mini-course is to illustrate a proof of Furstenberg’s ×2,×3 theorem: The ×2,×3 orbit of any irrational number on the unit interval is dense. Key results that will be needed for the proof are topological properties of irrational rotation on the unit interval. We will discuss those results and provide detailed backgrounds as well as proofs. At the end of the course, I will introduce various results and problems on digit expansions of integers. The following topics will be covered:
- Irrational rotations on torus;
- Diophantine approximation: Dirichlet theorem, Roth’s theorem, Baker’s theory of linear forms of logarithms;
- Furstenberg’s ×2,×3 theorem;
- Results and problems on digit expansions of integers;
- Furstenberg’s theorem on 2-dimensional torus (if time permits).
Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.
This video is part of the London Mathematical Society‘s Online Graduate Lecture Series. These are supported by the LMS.
