Doctoral School Mathematics and Scientific Computing

Abstract: Let ψ: {1,2,...}→[0,1/2] be given. Koukoulopoulos and Maynard (2020) proved the Duffin--Schaeffer conjecture: for almost all reals α there are infinitely many coprime solutions (p,q) to the inequality |α - p/q| < ψ(q)/q, if and only if the series Σ φ(q) ψ(q) / q is divergent. In a recent joint work with Christoph Aistleitner and Bence Borda, we established a quantitative version of this result in the following sense: for almost all α, the number of coprime solutions (p,q), subject to q <e Q, is of asymptotic order Ψ(Q) = Σ_q=1,...,Q 2φ(q) ψ(q) / q. In this talk, I will sketch the main steps of the original proof of Koukoulopoulos and Maynard as well as the additional ideas we used to obtain this quantification.[hide abstract]