Program
Renormalization Approach to the Littlewood Conjecture
Student No.:50
Time:Mon/Wed 13:30-15:05, 2017-07-3~2017-07-31 (no class on 07-24)
Instructor:Yitwah Cheung  [San Francisco State University]
Place:Conference Room 4, floor 2, Jin Chun Yuan West Building
Starting Date:2017-7-3
Ending Date:2017-7-31
 

Description:

 

The Littlewood Conjecture asks whether the triple product n|n*alpha-a||n*beta-b| can be made arbitrarily small for any pair of real numbers alpha and beta provided a,b and n are restricted to integers with n positive.

 
One reason that has been suggested for why the Conjecture has been so persistent is the absence of a suitable generalization of the all powerful Gauss continued fraction algorithm that harmoniously unites geometry with dynamics. Two of the most successful among the dozen or so that have been proposed are Rauzy induction and Minkowski continued fraction algorithm. The dynamical properties of the former are well-understood due to its connection to Teichmuller flows, but it has little relevance to Diophantine Approximation; the latter is more basic from the geometric viewpoint, and as such, more directly tied to number theory, but its renormalization dynamics is poorly understood. 
 

This course will illustrate the renormalization approach by developing several reformulations of the Littlewood Conjecture of a combinatorial nature. 

 

 

Prerequisite:

 

This course will be understandable to any student who is comfortable with the idea of a group action on a topological space.  Familiarity with the concept of measure would be helpful, but not essential.
 

 

 

Reference:

 

The literature on the Littlewood Conjecture is huge.  

 

A good starting point is the survey by Yann Bugeaud (Around the Littlewood conjecture in diophantine approximation, Publ. Math. Besançon Algèbr. Théor. Nr. (1) (2014), 5–18.)  Background for the reformulations of the Littlewood Conjecture can be found in the SFSU Master's Theses of S. Lui, D. Damon, T. Landry and L. Odom, which are available at http://math.sfsu.edu/cheung/index.html