Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$.

If we assume that $\alpha$ is irrational, then there exists an increasing sequence of integers $(n_k)_{k \in \mathbb{N}}$ such that $\|n_k \alpha - \beta\| \to 0$. Is it possible that the rate of convergence is exponential, that is, that there exists some $\eta > 1$ such that $\eta^{n_k} \|n_k \alpha - \beta\| \to 0$?

I believe that the fact that the sequence $\{n \alpha\}_{n \in \mathbb{N}}$ is equidistributed implies that if this is true for some $\eta > 1$, then it must be true for every $\eta > 1$, so this seems to be a rather strong condition.

If the rate may be exponential for general $\alpha$ and $\beta$, what about the case when $e^{2\pi i \alpha}$ and $e^{2\pi i \beta}$ are algebraic?