Duffin–Schaeffer theorem

The Koukoulopoulos–Maynard theorem, also known as the Duffin-Schaeffer conjecture, is a theorem in mathematics, specifically, the Diophantine approximation proposed as a conjecture by R. J. Duffin and A. C. Schaeffer in 1941 and proven in 2019 by Dimitris Koukoulopoulos and James Maynard. It states that if ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}}$ is a real-valued function taking on positive values, then for almost all ${\displaystyle \alpha }$ (with respect to Lebesgue measure), the inequality

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {f(q)}{q}}}$

has infinitely many solutions in coprime integers ${\displaystyle p,q}$ with ${\displaystyle q>0}$ if and only if

${\displaystyle \sum _{q=1}^{\infty }\varphi (q){\frac {f(q)}{q}}=\infty ,}$

where ${\displaystyle \varphi (q)}$ is Euler's totient function.

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.