Koukoulopoulos and Maynard solve the Duffin-Schaeffer Conjecture

Dimitris Koukoulopoulos (left) and James Maynard (right) after their announcement
(photo by K. Ford)

In July, 2019 at the Second Symposium on Analytic Number Theory in Cetraro, Italy, Dimitris Koukoulopoulos and James Maynard announced the solution of the Duffin-Schaeffer Conjecture (see also this article ), a 78-year old problem in Diophantine approximation concerned with which real numbers \(\alpha \) have infinitely many rational approximations of the form \[ \Big| \alpha - \frac{a}{b} \Big| < \frac{f(b)}{b}, \quad \text{gcd}(a,b)=1 \tag{1} \] for an arbitrary non-negative function \( f(b) \). The crux of the proof is to show that whenever the function \( f \) satisfies \[ \sum_{b=1}^\infty \frac{f(q)\phi(q)}{q} = \infty, \] where \( \phi \) is Euler's totient function, then the set of \( \alpha \in [0,1] \) so that (1) has infinitely solutions has full Lebesgue measure 1 (that is, the exceptional set has measure zero).