Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


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 "Khintchine’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.

share|improve this question
crossposted math.stackexchange.com/questions/427057/… –  Will Jagy Jun 22 '13 at 19:14
Have you taken a look at 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 20:06
Surely your university library can get it for you on interlibrary loan? –  Gerry Myerson Jun 23 '13 at 0:44
@AndresCaicedo - Thanks for the advice, but I didn't found any explicit mentioning of the theorem in the book (well, I don't have it, so I made a search on google books). The topics in the contents, too, not seem relevant. @GerryMyerson - Well, I'm not sure how long will it take or how much will it cost me. I'm not doing a research on the topic, so getting the article itself is not of utmost importance. –  user35222 Jun 23 '13 at 11:23
Ah, sorry, you are right! I pointed to the wrong book by Bugeaud (I've been looking at his two books recently). I should have said Approximation by algebraic numbers, 2004. Khintchine's theorem is section 1.3. –  Andres Caicedo Jun 26 '13 at 2:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.