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 α (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

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

Here ϕ(q) is the Euler totient function.

The full conjecture remains unsolved. However, the higher dimensional analogue of this conjecture has been resolved ^[2].

Progress

There have been many partial results of the Duffin-Schaeffer conjecture established to date. Paul Erdős established in 1970 that the conjecture holds if f(n) = c / n or f(n) = 0 for some positive constant c > 0. This was strengthened by Jeffrey Vaaler in 1978 to the case f(n) = O(n ^{− 1}) (http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf). More recently, this was strengthened to the conjecture being true whenever there exists some ε > 0 such that the series $\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \phi(n) = \infty$. This was done by Haynes, Pollington, and Velani in ^[3]

In 2006, Beresnevich and Velani proved that a Hausdorff dimension 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 ^[4]. Their result is available on the arXiv at http://arxiv.org/abs/math/0412141.

Notes

1. ^ R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Mathematical Journal, 8 (1941), 243–255
2. ^ A.D. Pollington and R.C. Vaughan, The k dimensional Duffin–Schaeffer conjecture, Mathematika, 37 (1990), 190–200
3. ^ A. Haynes, A. Pollington, and S. Velani, The Duffin-Schaeffer Conjecture with extra divergence, arXiv, (2009), http://arxiv.org/abs/0811.1234
4. ^ V. Beresnevich and S. Velani, A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Annals of Mathematics, 164 (2006), 971–992