Hermann Weyl Lectures

Given any non-negative function $\f:\mathbb{Z}\rightarrow\mathbb{R}$, it follows from basic ergodic ideas that either 100% of real numbers $\alpha$ have infinitely many rational approximations $a/q$ with $a,q$ coprime and $|\alpha-a/q|<f(q)$, or 0% of real numbers have infinitely many such approximations. Duffin and Schaeffer conjectured a simple criterion to establish when the 100% case occurs, and when the 0% case occurs.

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.