That existence of the
rational approximations implies
divergence of the
series follows from the
Borel–Cantelli lemma. This was strengthened by Jeffrey Vaaler in 1978 to the case f(n) = O(n^{-1}). More recently, this was strengthened to the conjecture being true whenever there exists some \varepsilon > 0 such that the series :\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \varepsilon} \varphi(n) = \infty. 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. == See also ==