Hi,

I'm trying to get a hold of Khintchine theorem in metric Diophantine approximation. Right now I'm interested in the divergence condition, namely:

If $\sum_{q=1}^\infty\psi(q) = \infty > $ and $\psi$ monotonically decreasing then $\lambda(\phi-approx)^c)=0$ (when $\psi(q) = q\phi(q)$, and $\phi-approx$ is the set of all $\theta$ such that there is an infinite number of solutions to $\|\theta - \frac{p}{q}\| < \phi(q)$).

I'm having trouble finding an example for the necessity of the monotonicity condition (=given a function that is not, $\lambda(\phi-approx)=0$). After some searching, I've found it mentioned in an article "Khintchineâ€™s problem in metric Diophantine approximation" by R. J. Duffin, A. C. Schaeffer. Unfortunately I don't have access to it (and neither my university library). Maybe someone has a free access link to it or could explain it shortly?

Thanks in advance.

Distribution Modulo One and Diophantine Approximation, by Yann Bugeaud? Khintchine's results are treated in good detail, and additional references are provided. – Andres Caicedo Jun 22 '13 at 20:06Approximation by algebraic numbers, 2004. Khintchine's theorem is section 1.3. – Andres Caicedo Jun 26 '13 at 2:55