The KoukoulopoulosâÂÂMaynard theorem, historically known as the DuffinâÂÂSchaeffer conjecture, is a theorem in mathematics, specifically Diophantine approximation. It was 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 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 with if and only if
where is Euler's totient function.
A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.
That existence of the rational approximations implies divergence of the series follows from the BorelâÂÂCantelli lemma. The converse implication is the crux of the conjecture.
There have been many partial results of the DuffinâÂÂSchaeffer conjecture established to date. Paul Erdà Âs 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
This was done by Haynes, Pollington, and Velani.
In 2006, Beresnevich and Velani proved that a Hausdorff measure analogue of the DuffinâÂÂSchaeffer conjecture is equivalent to the original DuffinâÂÂSchaeffer conjecture, which is a priori weaker. This result was published in the Annals of Mathematics.