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Most standard summaries of the literature on irrationality measure simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving any indication of how large qualifies as "sufficiently large." It would occasionally be useful (for example, for my answer here) to be able to come up with explicit bounds on the size of $q$.

I don't have access to Salikhov's paper where the exponent $7.6063$ is obtained, but Hata's paper which obtained an exponent of $8.0161$ is online here. Looking through it, the proof seems to be effective (all the bounds he's using appear to be defined explicitly), but it is definitely not written to be clear about what those bounds are.

Has anyone extracted explicit bounds on the size of the denominator from any irrationality measure calculation for $\pi$ more recent than Mahler's proof (referenced in the Hata paper) that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{42}} $$ unrestrictedly?

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  • $\begingroup$ I know this is random but by any chance did you go to New College? $\endgroup$
    – muaddib
    Jun 7, 2015 at 18:03
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    $\begingroup$ @muaddib: Nope, you must be thinking of some other Micah... $\endgroup$
    – Micah
    Jun 7, 2015 at 18:04
  • $\begingroup$ The exponent -42 got by Mahler was quite extraordinary in his time ("Striking inequality", Alan Baker). It is valid for all rational p/q with q > 1 and also indicates that rationals "may not be very closed to $\pi$". I guess all improvement of the exponent -42 is valid just "for all sufficiently large q" in the ordinary meaning of this expression without explicity giving of bounds. Maybe possible but certainly quite difficult to get them. $\endgroup$
    – Piquito
    Jun 7, 2015 at 18:51
  • $\begingroup$ I wasn't aware this was migrated, I'll delete my answer in that case. $\endgroup$ Jul 6, 2015 at 19:03
  • $\begingroup$ Salikhov's paper is available here: mathnet.ru/links/93c99ab6587bdc1bf0912ec96b1d9f4f/rm9175.pdf $\endgroup$ Apr 11, 2017 at 21:56

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