I'm interested in proving a possible generalization of Dirichlet's approximation theorem, where all integers involved are odd. The approximation theorem states that, for any irrational number $\alpha$, there are infinitely many integer pairs $(p,q)$ obeying: $$\left|p-q\alpha\right|<\frac{1}{q}$$
This is equivalent to saying $|p/q-\alpha| < 1/q^2$. In particular, the convergents $p/q$ of $\alpha$ satisfy this inequality, and moreover, at least every other convergent has odd denominator, so there are infinitely many pairs where $q$ is odd. Is it possible to prove a similar result where $p,q$ are restricted to be odd? I would accept any result which approximates $|p/q-\alpha|$ within $\mathcal{O}(1/q^2)$. Equivalently, my question is the following.
For every irrational $\alpha$, can we find infinitely many pairs of odd integers $p_n, q_n$ such that $|p_n-q_n\alpha| = \mathcal{O}(1/q_n)$?
I've been studying various proofs of the approximation theorem to try to get a generalization, but no luck yet. There might be some insight in the equidistribution theorem, but I haven't been able to figure anything out yet. A positive answer to this question would complete this partial answer I posted yesterday. For my purposes, I'm interested in the specific case where $\alpha=\frac{\pi}{2}$, hence $\alpha$ is transcendental, if that helps. I might also accept a slightly weaker result, like finding odd integers where $|p_n-q_n\alpha| = \mathcal{O}(q_n^{-1+\varepsilon})$ for some sufficiently small $\varepsilon>0$. It seems intuitively obvious that something like this should work, but I couldn't figure it out.
