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.