Duffin–Schaeffer theorem

Short description: Mathematical theorem

The Duffin–Schaeffer theorem is a theorem in mathematics, specifically, the Diophantine approximation proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if [math]\displaystyle{ f : \mathbb{N} \rightarrow \mathbb{R}^+ }[/math] is a real-valued function taking on positive values, then for almost all [math]\displaystyle{ \alpha }[/math] (with respect to Lebesgue measure), the inequality

[math]\displaystyle{ \left| \alpha - \frac{p}{q} \right| \lt \frac{f(q)}{q} }[/math]

has infinitely many solutions in coprime integers [math]\displaystyle{ p,q }[/math] with [math]\displaystyle{ q \gt 0 }[/math] if and only if

[math]\displaystyle{ \sum_{q=1}^\infty \varphi(q) \frac{f(q)}{q} = \infty, }[/math]

where [math]\displaystyle{ \varphi(q) }[/math] is Euler's totient function.

In 2019, the Duffin–Schaeffer conjecture was proved by Dimitris Koukoulopoulos and James Maynard.

Introduction

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 [math]\displaystyle{ c \gt 0 }[/math] such that for every integer [math]\displaystyle{ n }[/math] we have either [math]\displaystyle{ f(n) = c/n }[/math] or [math]\displaystyle{ f(n) = 0 }[/math]. This was strengthened by Jeffrey Vaaler in 1978 to the case [math]\displaystyle{ f(n) = O(n^{-1}) }[/math]. More recently, this was strengthened to the conjecture being true whenever there exists some [math]\displaystyle{ \varepsilon \gt 0 }[/math] such that the series

[math]\displaystyle{ \sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \varepsilon} \varphi(n) = \infty. }[/math]

This was done by Haynes, Pollington, and Velani.

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 was published in the Annals of Mathematics.

Proof

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.

See also

• Khinchin's theorem

References

• Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. 18. Oxford: Clarendon Press. ISBN 978-0-19-850083-4. 
• Harman, Glyn (2002). "One hundred years of normal numbers". in Bennett, M. A.; Berndt, B.C.; Boston, N. et al.. Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 57–74. ISBN 978-1-56881-162-8. 

External links

• Quanta magazine article about Duffin-Schaeffer conjecture.
• Numberphile interview with James Maynard about the proof.