A dynamical system proof of Niven's theorem and its extensions

Niven’s theorem asserts that $\{\cos (r\pi )\mid r\in \mathbb{Q}\}\cap \mathbb{Q}=\{0,\pm 1,\pm 1\mathrm{/}2\}\mathrm{.}$ In this paper, we use elementary techniques and results from arithmetic dynamics to obtain an algorithm for classifying all values in the set $\{\cos (r\pi )\mid r\in \mathbb{Q}\}\cap K$, where $K$ is an arbitrary number field.