Duffin–Schaeffer theorem

The Koukoulopoulos–Maynard theorem, also known as the Duffin-Schaeffer conjecture, 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 ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}}$ is a real-valued function taking on positive values, then for almost all ${\displaystyle \alpha }$ (with respect to Lebesgue measure), the inequality

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {f(q)}{q}}}$

has infinitely many solutions in coprime integers ${\displaystyle p,q}$ with ${\displaystyle q>0}$ if and only if

${\displaystyle \sum _{q=1}^{\infty }\varphi (q){\frac {f(q)}{q}}=\infty ,}$

where ${\displaystyle \varphi (q)}$ is Euler's totient function.

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.^[3][4][5]

Introduction

That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma.^[6] The converse implication is the crux of the conjecture.^[3] 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 ${\displaystyle c>0}$ such that for every integer ${\displaystyle n}$ we have either ${\displaystyle f(n)=c/n}$ or ${\displaystyle f(n)=0}$.^[3][7] This was strengthened by Jeffrey Vaaler in 1978 to the case ${\displaystyle f(n)=O(n^{-1})}$.^[8][9] More recently, this was strengthened to the conjecture being true whenever there exists some ${\displaystyle \varepsilon >0}$ such that the series

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {f(n)}{n}}\right)^{1+\varepsilon }\varphi (n)=\infty .}$

This was done by Haynes, Pollington, and Velani.^[10]

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.^[11]

See also

• Khinchin's theorem