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[1] and proven in 2019 by Dimitris Koukoulopoulos and James Maynard.
is a real-valued function taking on positive values, then for almost all
(with respect to Lebesgue measure), the inequality has infinitely many solutions in coprime integers
A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.
[3][4][5] That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma.
[6] The converse implication is the crux of the conjecture.
[3] There have been many partial results of the Duffin–Schaeffer conjecture established to date.
Paul Erdős established in 1970 that the conjecture holds if there exists a constant
[8][9] More recently, this was strengthened to the conjecture being true whenever there exists some
[10] In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker.