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

TL;DR

This paper establishes bounds for Diophantine exponents related to the action of $ ext{SL}_2( ext{ring of integers in a number field})$ on the complex plane, extending previous results from the integer case to more general rings.

Contribution

It provides new upper and lower bounds for Diophantine exponents of $ ext{SL}_2$ actions over discrete rings in $ ext{C}$, generalizing known results from the integer case.

Findings

Bounds are similar to Laurent and Nogueira's results for $ ext{SL}_2( ext{Z})$.

Uniform bounds are obtained outside a null measure set.

Results extend Diophantine approximation to complex number rings.

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 $\mathbb{C}^2 \setminus \{ \mathbf{0} \}$, 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 $\mathbb{R}^2 \setminus \{ \mathbf{0} \}$ albeit for us, uniformly nice bounds are obtained only outside of a set of null measure.