Define a rational number $\frac{p}{q}$ to be $\epsilon$-close to $\pi$ if
$$\left|\pi - \frac{p}{q}\right| \leq \frac{1}{q^{\mu-\epsilon}}$$
where $\mu$ is the irrationality measure of $\pi$, defined to be the largest real $\mu$ such that the above inequality has infinitely many solutions in positive integral $p, q$ where $\gcd(p, q) = 1$.
I'm trying to prove that there exist infinitely many $\epsilon$-close rational numbers to $\pi$ such that $p$ is even, and I can't figure out where to start.
How would I go about starting to prove this?
How hard would this be to prove (how difficult does this problem seem to people who know more about this field than I do?)
Is it likely that generalizations of this result are of a similar difficulty?
(more of a meta question) Is this the type of question that's better suited for MathOverflow?
I think the generalization I'd really like to prove is that for all positive integral $n\geq 2$ and positive integral $0<k<n, \gcd(k, n) = 1$ there exist infinitely $\epsilon$-close rational numbers to $\pi$ such that $p \equiv 0 \mod n$ and $q \equiv k \mod n$.