This is the second in a series of three, loosely connected talks. The previous talk in the series is here. The next talk in the series is here.
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.
This video is part of the Institute for Advanced Study‘s Hermann Weyl lecture series.
