# Recent progress in the irrationality measure of $\pi$

The context.

For any real number $$x$$, let's define the quantity

$$\mu(x):=\inf\left\{\mu\in\mathbb R_+\, \text{there is an infinity of rationals p/q such that}\ \left\vert x-\frac pq\right\vert<\frac 1{q^{\mu}}\right\},$$

and let's call it the irrationality measure of $$x$$.

We know that

$$\mu(\pi)\leqslant 8.016$$

thanks to M. Hata (1992).

The question.

We can read on this Wikipedia page that

$$\begin{equation} \mu(\pi)\leqslant 7.6063, \end{equation}$$

but this is provided without any reference.

1. Do you know any article where I could find a mention of this result?

2. Does this bound have been improved?

• Mathworld has a reference for the smaller bound: "Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008." I don't know if there's been any progress since 2008. – Micah May 12 '18 at 13:23
• @Arthur What $C$?? Assuming you mean the constant implied by $O()$ or $\ll$, then is not necessary in the definition of $\mu(x)$ because it can only affect the endpoint of the set $\{\mu \in \mathbb R_+ :\cdots\}$ and not the supremum. – Erick Wong May 12 '18 at 13:52
• @ErickWong Arthur was referring to a typo I made. I edited to correct it since. – E. Joseph May 12 '18 at 13:53
• @Micah: I don't believe a better value is currently known, because if there were a better value, then I'm sure it would turn up in the google search Salikhov + "on the irrationality measure of" (corresponding google scholar search). Also, this is the best value mentioned in Marko Leinonen's 2017 Ph.D. dissertation On Various Irrationality Measures. – Dave L. Renfro May 13 '18 at 17:55
• @DaveL.Renfro: Thanks! That sounds good enough for an answer to me. – Micah May 13 '18 at 18:04

## 1 Answer

Salikhov proved the smaller bound in: "Salikhov, V. Kh. "On the Irrationality Measure of pi." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008." as referenced, e.g., on Mathworld.

Marko Leinonen's 2017 Ph.D. dissertation On Various Irrationality Measures claims (at the bottom of page 13) that this is the currently best-known bound. So (as of when I'm writing this, May 2018) it almost certainly has not been superseded.