The Duffin-Schaeffer Conjecture

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|<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.