Duffin–Schaeffer conjecture
The Duffin–Schaeffer conjecture is a conjecture (now a theorem) in mathematics, specifically, the Diophantine approximation proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if formula_1 is a real-valued function taking on positive values, then for almost all formula_2 (with respect to Lebesgue measure), the inequality has infinitely many solutions in coprime integers formula_4 with formula_5 if and only if where formula_7 is Euler's totient function. In 2019, the Duffin–Schaeffer conjecture was proved by Dimitris Koukoulopoulos and James Maynard. Progress. That existence of the rational approximations implies divergence  of the series follows from the Borel–Cantelli lemma. The converse implication is the crux of the conjecture. 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 formula_8 such that for every integer formula_9 we have either formula_10 or formula_11. This was strengthened by Jeffrey Vaaler in 1978 to the case formula_12. More recently, this was strengthened to the conjecture being true whenever there exists some formula_13 such that the series 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. This result is published in the "Annals of Mathematics". In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture. In July 2020, the proof was published in the Annals of Mathematics. Related problems. A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.