Rotational partition function: Difference between revisions

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The '''Duffin–Schaeffer conjecture''' is an important conjecture in [[metric number theory]] proposed by [[Richard Duffin|R. J. Duffin]] and [[Albert Charles Schaeffer|A. C. Schaeffer]] in 1941.<ref>R. J. Duffin and A. C. Schaeffer, ''Khintchine's problem in metric [[Diophantine approximation]]'', [[Duke Mathematical Journal]], '''8''' (1941), 243–255</ref> It states that if <math>f : \mathbb{N} \rightarrow \mathbb{R}^+</math> is a real-valued function taking on positive values, then for [[almost all]] <math> \alpha </math> (with respect to [[Lebesgue measure]]), the inequality


: <math> \left | \alpha - \frac{p}{q} \right | < \frac{f(q)}{q} </math>


has infinitely many solutions in [[co-prime]] integers <math>p,q</math> with <math>q > 0</math> if and only if the sum
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: <math> \sum_{q=1}^\infty f(q) \frac{\varphi(q)}{q} = \infty. </math>
 
Here <math> \varphi(q)</math> is the [[Euler totient function]].
 
The full conjecture remains unsolved. However, a higher dimensional analogue of this conjecture has been resolved.<ref name=Mon204/><ref>{{cite journal | first1=A.D. | last1=Pollington | first2=R.C. | last2=Vaughan | author2-link=Bob Vaughan | title=The ''k'' dimensional Duffin–Schaeffer conjecture | journal=Mathematika | volume=37 | number=2 | year=1990 | pages=190–200 | zbl=0715.11036 | issn=0025-5793 }}</ref><ref name=Har0269>Harman (2002) p.69</ref>
 
==Progress==
The implication from the existence of the rational approximations to the divergence of the series follows from the [[Borel–Cantelli lemma]].<ref name=Har0268>Harman (2002) p.68</ref> The converse implication is the crux of the conjecture.<ref name=Mon204/>
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>c > 0 </math> such that for every integer <math> n </math> we have either <math>f(n) = c/n</math> or <math>f(n) = 0</math>.<ref name=Mon204>{{ cite book | last=Montgomery | first=Hugh L. | author-link=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 | page=204 }}</ref><ref name=Har27>Harman (1998) p.27</ref> This was strengthened by Jeffrey Vaaler in 1978 to the case <math>f(n) = O(n^{-1})</math>.<ref>http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf</ref><ref name=Har28>Harman (1998) p.28</ref>  More recently, this was strengthened to the conjecture being true whenever there exists some <math>\epsilon > 0 </math> such that the series <math>\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \varphi(n) = \infty </math>. This was done by Haynes, Pollington, and Velani.<ref>A. Haynes, A. Pollington, and S. Velani, ''The Duffin-Schaeffer Conjecture with extra divergence'', arXiv, (2009), http://arxiv.org/abs/0811.1234</ref>
 
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]]''.<ref>{{cite journal | last1=Beresnevich | first1=Victor | last2=Velani | first2=Sanju | title=A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures | journal=[[Annals of Mathematics]] (2) | volume=164 | number=3 | year=2006 | pages=971–992 | url=http://arxiv.org/abs/math/0412141 | zbl=1148.11033 | issn=0003-486X }}</ref>
 
==Notes==
{{Reflist}}
 
==References==
* {{cite book | last=Harman | first=Glyn | author-link= | title=Metric number theory | series=London Mathematical Society Monographs. New Series | volume=18 | location=Oxford | publisher=[[Clarendon Press]] | year=1998 | isbn=0-19-850083-1 | zbl=1081.11057 }}
* {{cite book | last=Harman | first=Glyn | authorlink=Glyn Harman | chapter=One hundred years of normal numbers | editor1-last=Bennett | editor1-first=M. A. | editor2-last=Berndt | editor2-first=B.C. | editor2-link=Bruce C. Berndt | editor3-last=Boston | editor3-first=N. | editor3-link=Nigel Boston | editor4-last=Diamond | editor4-first=H.G. | editor5-last=Hildebrand | editor5-first=A.J. | editor6-last=Philipp | editor6-first=W. | title=Surveys in number theory: Papers from the millennial conference on number theory | location=Natick, MA | publisher=A K Peters | pages=57–74 | year=2002 | isbn=1-56881-162-4 | zbl=1062.11052 }}
 
{{DEFAULTSORT:Duffin-Schaeffer conjecture}}
[[Category:Conjectures]]
[[Category:Diophantine approximation]]

Latest revision as of 10:34, 12 November 2014


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