Proving the Duffin-Schaeffer conjecture without GCD graphs

We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the ( (N) ) ^-C error-term of Aistleitner-Borda and the first named author to (- ( (N) ) ^1{2 - }).