In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $a_0 = a_1 = 1$ and $a_{m} = 2a_{m-1} - 3a_{m-2}$ satisfies $\lim_{m\to \infty} |a_m| = \infty$. It seems somewhat easy to prove at first glance (and the actual behavior of $|a_m|$ is exponential), one needs to deal with possible cancellation. The general term is given by $$ a_m = \frac{1}{2} ((1 + \sqrt{-2})^{m} + (1 - \sqrt{-2})^{m}) = \sqrt{3}^m \cos (m \alpha) $$ where $\alpha = \arctan(\sqrt{2})$, and if $m\alpha$ is sufficiently close to the odd multiple of $\pi / 2$, then $\cos(m\alpha)$ could be very close to zero. So one may need to show that $\cos(m\alpha)$ can't be exponentially small with respect to $m$ in some sense, or use $p$-adic method as in Conrad's note.
What I thought is that once we know $\lim_{m\to\infty}|a_m| = \infty$, this would tell us that $\alpha / \pi$ can't be very close to rational numbers, and may tell something about irrationality measure of $\alpha / \pi$. I tried some but didn't get anything useful at this moment. Is it possible to deduce some information on the irrationality measure of $\alpha /\pi$ using divergence, at least finiteness or infiniteness?