Duffin–Schaeffer conjecture

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 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

\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 the sum

\sum_{q=1}^\infty f(q) \frac{\varphi(q)}{q} = \infty.

Here \varphi(q) is the Euler totient function.

The full conjecture remains unsolved. However, a higher-dimensional analogue of this conjecture has been resolved.[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 c > 0 such that for every integer n we have either f(n) = c/n or f(n) = 0.[2][6] This was strengthened by Jeffrey Vaaler in 1978 to the case f(n) = O(n^{-1}).[7][8] More recently, this was strengthened to the conjecture being true whenever there exists some \epsilon > 0 such that the series \sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \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]

Notes

  1. ↑ Duffin, R. J.; Schaeffer, A. C. (1941). "Khintchine’s problem in metric diophantine approximation". Duke math. J. 8: 243–255. doi:10.1215/S0012-7094-41-00818-9. JFM 67.0145.03. Zbl 0025.11002.
  2. ↑ 2.0 2.1 2.2 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 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. 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. ↑ 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. doi:10.4007/annals.2006.164.971. ISSN 0003-486X. Zbl 1148.11033.

References