Submit progress

on Irrationality measure of $\pi$ ($C_{7a}$)

The statement

We define $C_{7a}$ to be the irrationality measure of $\pi$ :

$C_{7a} := \sup_{\mu\in\mathbb{R}} \mu$ such that $\lvert \pi - p/q \rvert < q^{-\mu}$ for infinitely many rationals $p/q$ .

Equivalently, $C_{7a}$ is the infimum of all $\nu$ such that for every $\varepsilon>0$ there exists $q_{0}(\varepsilon)$ with

\left|\pi-\frac{p}{q}\right| > \frac{1}{q^{\nu+\varepsilon}}

for all integers $p$ and all integers $q \ge q_{0}(\varepsilon)$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.