I'm very curious about the relationship between $\pi$ and irrational numbers in general. Whether $\pi$ contains all possible number combinations is unknown, but expected to be true.

What "kind" of irrationals numbers does $\pi$ contain? Consider the set $$ S=\{\alpha\pi+\beta\,\mid\,\alpha,\beta\in\mathbb{Q},\,\alpha\neq 0\}. $$ Since $\pi$ contains only countably many infinite digit strings, we have that $S\subsetneq\mathbb{R}\backslash\mathbb{Q}$. What else can we say about $S$?

For example, one may ask the following questions:

- Is it true that $\sqrt{2}\in S$? If not, what prevents it from being true?
- What about $e$? Are $S$ and transcendental numbers relatable somehow?

Any deeper insights are appreciated.