Video Lectures

Given any non-negative function \f:ℤ→ℝ, it follows from basic ergodic ideas that either 100% of real numbers α have infinitely many rational approximations a/q with a,q coprime and |α−a/q|

 

I'll describe a recent resolution of this conjecture, which recasts the problem in combinatorial language, and then uses a general 'structure vs randomness' principle combined with an iterative argument to solve this combinatorial problem.

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| \gg \min \{ |A|^{1+c}, |G| \}. In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset...