If the 'optimal' Diophantine approximation of $\pi$ is given by the maximum value of $M=-\log_q(\min_{\forall p \in \mathbb{N}} |\frac{p}{q}-\pi|)$ for $q \geq 2$, what is this value?
$\begingroup$
$\endgroup$
4
-
2$\begingroup$ mathoverflow.net/questions/210509/… $\endgroup$– Felipe VolochCommented Jul 8, 2016 at 6:13
-
8$\begingroup$ Not the same question. The answer here is almost surely the $3.429288+$ obtained from the approximation $22/7$, though $355/113$ is an impressive also-ran at about $3.202$ (further approximations should converge rapidly to $2$, but we have no technique for proving this). $\endgroup$– Noam D. ElkiesCommented Jul 8, 2016 at 6:31
-
$\begingroup$ Interesting. Are there any papers on the subject that you would recommend? $\endgroup$– Aidan RockeCommented Jul 8, 2016 at 6:34
-
$\begingroup$ Recently, I encountered a relevant problem: whether or not there are integers $p, q$ with $q > 2$ such that $|\pi - \frac{p}{q}| < \frac{1}{q^4}$? $\endgroup$– River LiCommented Dec 28, 2019 at 2:18
Add a comment
|