Duffin–Schaeffer conjecture


The Duffin–Schaeffer conjecture is an important conjecture in mathematics, specifically metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if is a real-valued function taking on positive values, then for almost all , the inequality
has infinitely many solutions in co-prime integers with if and only if
where is the Euler totient function.
A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.

Progress

The implication from the existence of the rational approximations to the 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 such that for every integer we have either or. This was strengthened by Jeffrey Vaaler in 1978 to the case. More recently, this was strengthened to the conjecture being true whenever there exists some 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.