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. 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

• Harman, Glyn (1998). Metric number theory. London Mathematical Society Monographs. New Series 18. Oxford: Clarendon Press. ISBN 0-19-850083-1. 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. Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 57–74. ISBN 1-56881-162-4. Zbl 1062.11052.