Duffin–Schaeffer conjecture

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

The Duffin–Schaeffer conjecture is an important conjecture in metric number theory 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 co-prime integers with if and only if

where is the Euler totient function.

A higher-dimensional analogue of this conjecture was resolved by Vaughan and Pollington in 1990.[2][3][4]


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 such that for every integer we have either or .[2][6] This was strengthened by Jeffrey Vaaler in 1978 to the case .[7][8] 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.[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]


  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. ^ a b c 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.
  3. ^ 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.
  4. ^ Harman (2002) p. 69
  5. ^ Harman (2002) p. 68
  6. ^ Harman (1998) p. 27
  7. ^ "Department of Mathematics" (PDF).
  8. ^ Harman (1998) p. 28
  9. ^ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), https://arxiv.org/abs/0811.1234
  10. ^ 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.
  11. ^ Koukoulopoulos, D.; Maynard, J. (2019). "On the Duffin–Schaeffer conjecture". arXiv:1907.04593.


External links[edit]