To address the question of explicit bounds on the denominator ( q ) for irrationality measures of ( \pi ) with exponents like 7.6063 (Salikhov) or 8.0161 (Hata), we first recall that the irrationality measure ( \mu(\pi) ) quantifies how well ( \pi ) can be approximated by rationals ( p/q ). Recent results (post-Mahler) provide sharper exponents but typically state them asymptotically ("f...

Short answer: Yes in principle, but in practice no one seems to have published a clean, usable value of the “sufficiently large” threshold for the best-known exponents for π; extracting it from the proofs is possible but leads to astronomically large numbers, making it of no practical use. The only widely available completely explicit inequality that holds for all q is still Mahler’s old result (w...

This is an excellent question about the effectiveness of irrationality measures for π. The gap between theoretical results and practical bounds is indeed a significant issue in Diophantine approximation.

Current State of Knowledge

The best known irrationality measure for π is indeed Salikhov's result from 2008, which gives μ(π) < 7.6063. However, as you correctly note, these results are typic...

Yes – if one is willing to “open all the boxes’’ in the recent proofs the lower bounds can be written in the completely explicit form

    |π – p / q|  ≥  C · q –μ      (  q  ≥  q0 ).                   (1)

Both Hata’s exponent μ = 8.0161… (1993) and Salikhov’s record μ = 7.6063… (2008) can be made effective; the constants turn out to be absurdly large but they are perfectly explicit.
Belo...

Yes – all the modern proofs that give an upper bound for the irrationality exponent of π are completely effective; one can in principle (and in practice) write down a concrete constant c > 0 and a concrete size q₀ such that

  |π − p/q| > c q^{–μ} (1)

holds for every pair of coprime integers p,q with q ≥ q₀. This has actually been done in print fo...

The current best irrationality measure for π is about 7.6063, from Salikhov, but I don't have access to t...

Here is what is known in the literature about effective irrationality‐measures of π and the only place where one actually finds a fully explicit “for all q ≥ q₀” stated with a concrete q₀. Beyond Mahler’s original result (exponent 42) no one has ever written down an explicit threshold q₀ for any of the later, sharper exponents.

1. Mahler (1953)
   • He showed unconditionally that
   |...

Excellent question. You've pinpointed a common frustration in number theory, where the main results are often stated asymptotically, leaving the "effective" part as a difficult and often un-published exercise. The work to compute the explicit threshold is highly non-trivial and requires careful effective versions of all analytic estimates used in the proof.

Fortunately, there have been effo...

Effective Irrationality Measures for π

You're asking about explicit bounds in irrationality measure results for π - specifically, you want to know how large "sufficiently large" is in the inequality:

This is indeed a detail often omitted in summaries of irrationality measure results.

Current state of knowledge

The most recent eff...