Let $e^{i\phi}, e^{i\theta} \in \mathbb{S}^1$. Let $p(x) = x + 2k\pi$, where $k \in \mathbb{Z}$ is chosen so $p(x) \in [0, 2\pi)$.
If we assume that $\phi/\pi$ is irrational, then there exists an increasing sequence of integers $(n_k)_{k \in \mathbb{N}}$ such that $|p(n_k\phi) - p(\theta)| \to 0$. Is it possible that the rate of convergence is exponential, that is, does there exist some $\eta > 1$ such that $\eta^{n_k}|p(n_k\phi) - p(\theta)| \to 0$?
I believe Baker's theorem may be useful in the case that $e^{i\phi}$ and $e^{i\theta}$ are algebraic and $p(\theta) \neq p(q \phi)$ for every $q \in \mathbb{Q}$, but I'm not sure if works. I also believe that the fact that $(p(n_k \phi))_k$ is equidistributed on $[0, 2\pi)$ may be useful, since intuitively it implies that if $\eta^{n_k}|p(n_k\phi) - p(\theta)| \to 0$ for some $\eta > 1$, then $\eta^{n_k}|p(n_k\phi) - p(\theta)| \to 0$ for every $\eta > 1$.