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 such that for every
integer we have either or .[3][7] This was strengthened by Jeffrey Vaaler in 1978 to the case .[8][9] 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.[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]
Harman, Glyn (2002). "One hundred years of normal numbers". In Bennett, M. A.;
Berndt, B.C.;
Boston, N.; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. (eds.). Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 57–74.
ISBN978-1-56881-162-8.
Zbl1062.11052.