Duffin–Schaeffer_theorem
Duffin–Schaeffer theorem
Mathematical theorem
The Duffin–Schaeffer theorem 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.[2] 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.[3][4][5]