Duffin–Schaeffer theorem
The Duffin–Schaeffer theorem is a theorem in mathematics, specifically, the Diophantine approximation proposed by R. J. Duffin and A. C. Schaeffer in 1941.[1] It states that if [math]\displaystyle{ f : \mathbb{N} \rightarrow \mathbb{R}^+ }[/math] is a real-valued function taking on positive values, then for almost all [math]\displaystyle{ \alpha }[/math] (with respect to Lebesgue measure), the inequality
- [math]\displaystyle{ \left| \alpha - \frac{p}{q} \right| \lt \frac{f(q)}{q} }[/math]
has infinitely many solutions in coprime integers [math]\displaystyle{ p,q }[/math] with [math]\displaystyle{ q \gt 0 }[/math] if and only if
- [math]\displaystyle{ \sum_{q=1}^\infty \varphi(q) \frac{f(q)}{q} = \infty, }[/math]
where [math]\displaystyle{ \varphi(q) }[/math] is Euler's totient function.
In 2019, the Duffin–Schaeffer conjecture was proved by Dimitris Koukoulopoulos and James Maynard.[2]
Introduction
That existence of the rational approximations implies divergence of the series follows from the Borel–Cantelli lemma.[3] The converse implication is the crux of the conjecture.[4] 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]\displaystyle{ c \gt 0 }[/math] such that for every integer [math]\displaystyle{ n }[/math] we have either [math]\displaystyle{ f(n) = c/n }[/math] or [math]\displaystyle{ f(n) = 0 }[/math].[4][5] This was strengthened by Jeffrey Vaaler in 1978 to the case [math]\displaystyle{ f(n) = O(n^{-1}) }[/math].[6][7] More recently, this was strengthened to the conjecture being true whenever there exists some [math]\displaystyle{ \varepsilon \gt 0 }[/math] such that the series
- [math]\displaystyle{ \sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \varepsilon} \varphi(n) = \infty. }[/math]
This was done by Haynes, Pollington, and Velani.[8]
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.[9]
Proof
In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture.[10][11] In July 2020, the proof was published in the Annals of Mathematics.[2]
Related problems
A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.[4][12][13]
See also
Notes
- ↑ Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine's problem in metric diophantine approximation". Duke Math. J. 8 (2): 243–255. doi:10.1215/S0012-7094-41-00818-9.
- ↑ 2.0 2.1 Koukoulopoulos, Dimitris; Maynard, James (2020). "On the Duffin-Schaeffer conjecture". Annals of Mathematics 192 (1): 251. doi:10.4007/annals.2020.192.1.5. https://www.jstor.org/stable/10.4007/annals.2020.192.1.5.
- ↑ Harman (2002) p. 68
- ↑ 4.0 4.1 4.2 Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. 84. Providence, RI: American Mathematical Society. p. 204. ISBN 978-0-8218-0737-8.
- ↑ Harman (1998) p. 27
- ↑ "Duffin-Schaeffer Conjecture". 2010-08-09. https://math.osu.edu/sites/math.osu.edu/files/duffin-schaeffer-conjecture.pdf.
- ↑ Harman (1998) p. 28
- ↑ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), https://arxiv.org/abs/0811.1234
- ↑ Beresnevich, Victor; Velani, Sanju (2006). "A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures". Annals of Mathematics. Second Series 164 (3): 971–992. doi:10.4007/annals.2006.164.971. ISSN 0003-486X.
- ↑ Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593 [math.NT].
- ↑ Sloman, Leila (2019). "New Proof Solves 80-Year-Old Irrational Number Problem". Scientific American. https://www.scientificamerican.com/article/new-proof-solves-80-year-old-irrational-number-problem/.
- ↑ Pollington, A.D.; Vaughan, R.C. (1990). "The k dimensional Duffin–Schaeffer conjecture". Mathematika 37 (2): 190–200. doi:10.1112/s0025579300012900. ISSN 0025-5793. http://www.numdam.org/item/JTNB_1989__1_1_81_0/.
- ↑ Harman (2002) p. 69
References
- Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. 18. Oxford: Clarendon Press. ISBN 978-0-19-850083-4.
- Harman, Glyn (2002). "One hundred years of normal numbers". in Bennett, M. A.; Berndt, B.C.; Boston, N. et al.. Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 57–74. ISBN 978-1-56881-162-8.
External links
- Quanta magazine article about Duffin-Schaeffer conjecture.
- Numberphile interview with James Maynard about the proof.
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