Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$


According to [this document](https://math.osu.edu/sites/math.osu.edu/files/What_is_2018_Markov_Lagrange_Spectra.pdf), if we prove that $\pi$ has unbounded coefficients in its continued fraction expansion, then the answer to the above question is affirmative.



Markov constant of $\pi$
I believe that this is an open problem. See e.g. the bottom of page 202 of [this article](https://www.maa.org/sites/default/files/pdf/pubs/BaileyBorweinPiDay.pdf) by Bailey and Borwein. BTW the question was asked here [before](https://mathoverflow.net/questions/99300/is-pi-well-approximable).


