Is it known whether the language $L_π = \{w\in\{0,1\}^* : w\text{ appears in the binary expansion of }π\}$ is decidable?
- $L_π$ is easily recognizable (a.k.a. computably enumerable).
- A trivially decidable very classical variant of the language is $\{n:0^n\text{ appears in the binary expansion of }π\}$.
- There is a conjecture that $π$ is disjunctive in base 2, that is its binary expansion contains all possible finite strings: It implies decidability since in such case, $L_π = \{0,1\}^*$.
So a detailed version of my question:
- Can we prove decidability without assuming the conjecture?
- Does decidability implies something about the conjecture?
- Do we have similar results for other well-known irrational constants?
