Convergents of Simultaneous Approximation
Student No.:40
Time:Tue & Thu 13:30-15:05, Jul.17-Aug.16
Instructor:张翼华 Yitwah Cheung  
Place:Conference Room 3, Jin Chun Yuan West Bldg.
Starting Date:2018-7-17
Ending Date:2018-8-16

The Littlewood Conjecture is a long-standing open problem since the 1930s about the simultaneous approximation of a pair of real numbers by rationals. It is implied by Margulis Conjecture which gives a nearly equivalent reformulation from a dynamical point of view in terms of a rank 2 action on the space of lattices. The set of counterexamples is known to have Hausdorff dimension zero by a celebrated result of Einsiedler-Katok-Lindenstrauss in 2006 based on the idea of entropy and measure rigidity. The best known "near counterexamples" are due to Badziahin in 2011. In this lecture series, I will explain a new approach to the Littlewood conjecture based on generalizing the idea of a convergent of the continued fraction expansion, introducing ideas of renormalization and giving an elementary exposition of the dynamical reformulation and motivating the connections with other models such as the geodesic flow on the modular surface and providing by analog an entry point for the study of Teichmuller flows on space of holomorphic differentials on Riemann surfaces.


Familiarity with basic notions from Topology (at the level of Munkres) and the rudiments of Lebesgue theory.


[1] Continued Fractions, by A. Ya. Khinchin (Dover Books);
[2] Exploring Number Jungle: a journel into Diophantine Analysis, by Edward B. Burger (AMS SML 8);
[3] Ergodic Theory with a view towards Number Theory by M. Einsiedler and T. Ward (GTM 259).