Duffin–Schaeffer conjecture

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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 the sum

Here is the Euler totient function.

The full conjecture remains unsolved. However, a higher-dimensional analogue of this conjecture has been resolved.[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]


  1. ^ Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine’s problem in metric diophantine approximation". Duke math. J. 8: 243–255. JFM 67.0145.03. Zbl 0025.11002. doi:10.1215/S0012-7094-41-00818-9. 
  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 0-8218-0737-4. Zbl 0814.11001. 
  3. ^ Pollington, A.D.; Vaughan, R.C. (1990). "The k dimensional Duffin–Schaeffer conjecture". Mathematika. 37 (2): 190–200. ISSN 0025-5793. Zbl 0715.11036. doi:10.1112/s0025579300012900. 
  4. ^ Harman (2002) p.69
  5. ^ Harman (2002) p.68
  6. ^ Harman (1998) p.27
  7. ^ http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf
  8. ^ Harman (1998) p.28
  9. ^ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), http://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 (2). 164 (3): 971–992. ISSN 0003-486X. Zbl 1148.11033. doi:10.4007/annals.2006.164.971.