The Duffin–Schaeffer conjecture is an important conjecture in mathematics, specifically metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941.[1] It states that if \( {\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}} \) is a real-valued function taking on positive values, then for almost all \( \alpha \) (with respect to Lebesgue measure), the inequality
\( {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {f(q)}{q}}} \)
has infinitely many solutions in co-prime integers p,q with q>0 if and only if
\( {\displaystyle \sum _{q=1}^{\infty }f(q){\frac {\varphi (q)}{q}}=\infty ,} \)
where φ ( q ) {\displaystyle \varphi (q)} {\displaystyle \varphi (q)} is the Euler totient function.
A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.[2][3][4]
Progress
The implication from the existence of the rational approximations to the divergence of the series follows from the Borel–Cantelli lemma.[5] The converse implication is the crux of the conjecture.[2] 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 \({\displaystyle c>0} \) such that for every integer n we have either \( {\displaystyle f(n)=c/n} \) or f(n)=0.[2][6] This was strengthened by Jeffrey Vaaler in 1978 to the case \( {\displaystyle f(n)=O(n^{-1})} \) .[7][8] More recently, this was strengthened to the conjecture being true whenever there exists some \( \varepsilon >0 \) such that the series
\( {\displaystyle \sum _{n=1}^{\infty }\left({\frac {f(n)}{n}}\right)^{1+\varepsilon }\varphi (n)=\infty } \) . This was done by Haynes, Pollington, and Velani.[9]
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 is published in the Annals of Mathematics.[10]
In July 2019, Dimitris Koukoulopoulos and James Maynard announced a proof of the conjecture.[11][12][13]
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. JFM 67.0145.03. Zbl 0025.11002.
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. Zbl 0814.11001.
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. Zbl 0715.11036.
Harman (2002) p. 69
Harman (2002) p. 68
Harman (1998) p. 27
"Duffin-Schaeffer Conjecture" (PDF). Ohio State University Department of Mathematics. 2010-08-09. Retrieved 2019-09-19.
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. arXiv:math/0412141. doi:10.4007/annals.2006.164.971. ISSN 0003-486X. Zbl 1148.11033.
Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593.
Sloman, Leila (2019). "New Proof Solves 80-Year-Old Irrational Number Problem". Scientific American.
https://www.youtube.com/watch?v=1LoSV1sjZFI
References
Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. 18. Oxford: Clarendon Press. ISBN 978-0-19-850083-4. Zbl 1081.11057.
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. ISBN 978-1-56881-162-8. Zbl 1062.11052.
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