Height limits, equidistribution, and dilogarithms

Abstract

When $R$ is a rational (or even algebraic) function, it is well known how the height of $R(\alpha )$ behaves asymptotically as the height of $\alpha$ tends to $\infty$. Here we prove some analogous versions as the height of $\alpha$ tends instead to $0$, also with explicit error terms. Our most general results, some with a trigonometrical flavour, are for roots of unity of large order. For example, the height of $\tan 2\pi /n$ tends to $2G/\pi$ where $G$ is the Catalan constant.