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.

  • $\begingroup$ crossposted math.stackexchange.com/questions/427057/… $\endgroup$
    – Will Jagy
    Jun 22, 2013 at 19:14
  • $\begingroup$ 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. $\endgroup$ Jun 22, 2013 at 20:06
  • $\begingroup$ Surely your university library can get it for you on interlibrary loan? $\endgroup$ Jun 23, 2013 at 0:44
  • $\begingroup$ @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. $\endgroup$
    – user35222
    Jun 23, 2013 at 11:23
  • $\begingroup$ 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. $\endgroup$ Jun 26, 2013 at 2:55

1 Answer 1


you can see the proof of Khintchine's theorem in "Diophantine approximation by W.S. Cassels" or in relation to contributions towards Duffin-Schaeffer conjecture see "Metric Number Theory by Glynn Harman". However the theorem can easily be prove as a consequence of 'ubiquity' framework introduced in "Measure theoretic laws for limsup sets by V. Beresnevich, D. Dickinson and S. Valani".

  • $\begingroup$ I agree. The standard resource for all questions in metric Diophantine approximation is Harman's book on Metric Number Theory. $\endgroup$ Feb 3, 2015 at 9:06

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