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 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 coprime integers with if and only if

where 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 such that for every integer we have either or .[4][5] This was strengthened by Jeffrey Vaaler in 1978 to the case .[6][7] 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.[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]

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.[4][12][13]

See also

Notes
  1. 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.
  2. Koukoulopoulos, Dimitris; Maynard, James (2020). "On the Duffin-Schaeffer conjecture". Annals of Mathematics. 192 (1): 251. arXiv:1907.04593. doi:10.4007/annals.2020.192.1.5. JSTOR 10.4007/annals.2020.192.1.5. S2CID 195874052.
  3. Harman (2002) p. 68
  4. Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. p. 204. ISBN 978-0-8218-0737-8. Zbl 0814.11001.
  5. Harman (1998) p. 27
  6. "Duffin-Schaeffer Conjecture" (PDF). Ohio State University Department of Mathematics. 2010-08-09. Retrieved 2019-09-19.
  7. Harman (1998) p. 28
  8. A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), https://arxiv.org/abs/0811.1234
  9. 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. S2CID 14475449. Zbl 1148.11033.
  10. Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593 [math.NT].
  11. Sloman, Leila (2019). "New Proof Solves 80-Year-Old Irrational Number Problem". Scientific American.
  12. 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. S2CID 122789762. Zbl 0715.11036.
  13. Harman (2002) p. 69

References
  • Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series. Vol. 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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