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

Abstract

Niven's theorem asserts that $\{\cos(r\pi) \mid r\in \mathbb{Q}\}\cap \mathbb{Q}=\{0,\pm 1,\pm 1/2\}.$ 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.