Duffin–Schaeffer conjecture
Encyclopedia
The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941 . 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 co-prime integers
with 
if and only if the sum
Here
is the Euler totient function.
The full conjecture remains unsolved. However, the higher dimensional analogue of this conjecture has been resolved .
established in 1970 that the conjecture holds if
or 
for some positive constant 
. This was strengthened by Jeffrey Vaaler in 1978 to the case 
(http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf). 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 in
In 2006, Beresnevich and Velani proved that a Hausdorff dimension analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin-Schaeffer conjecture, which is a priori weaker. This result is published in the Annals of Mathematics
. Their result is available on the arXiv
at http://arxiv.org/abs/math/0412141.
is a real-valued function taking on positive values, then for almost allAlmost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
(with respect to Lebesgue measureLebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
), the inequality
has infinitely many solutions in co-prime integers
with if and only if the sum
Here
is the Euler totient function.The full conjecture remains unsolved. However, the higher dimensional analogue of this conjecture has been resolved .
Progress
There have been many partial results of the Duffin-Schaeffer conjecture established to date. Paul ErdősPaul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
established in 1970 that the conjecture holds if
or for some positive constant . This was strengthened by Jeffrey Vaaler in 1978 to the case (http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf). 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 in In 2006, Beresnevich and Velani proved that a Hausdorff dimension analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin-Schaeffer conjecture, which is a priori weaker. This result is published in the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
. Their result is available on the arXiv
ArXiv
The arXiv |Chi]], χ) is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance which can be accessed online. In many fields of mathematics and physics, almost all...
at http://arxiv.org/abs/math/0412141.