Diophantine exponents for standard linear actions of ${\mathrm{SL}}_2$ over discrete rings in $\mathbb{C}$

Abstract

We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of ${\mathrm{SL}}_2({\mathcal{O}}_K)$ on the punctured complex plane ${\mathbb{C}}^2\setminus \{0\}$, where $K$ is a number field whose ring of integers ${\mathcal{O}}_K$ is discrete and within a unit distance of any complex number. The results are similar to those of Laurent and Nogueira for ${\mathrm{SL}}_2(\mathbb{Z})$ action on ${\mathbb{R}}^2\setminus \{0\}$ albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.

arXiv comment16 pages