Diophantine exponents for standard linear actions of ${\rm SL}_2$ over discrete rings in $\mathbb {C}$
Volume 177 / 2017
Acta Arithmetica 177 (2017), 53-73
MSC: Primary 11J20, 11J13, 11A55; Secondary 22Fxx.
DOI: 10.4064/aa8370-6-2016
Published online: 23 December 2016
Abstract
We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of ${\mathrm{SL}_{2}( \mathcal 0_K )}$ 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 any complex number is within a unit distance of some element of $\mathcal O_K$. The results are similar to those of Laurent and Nogueira (2012) for the ${\mathrm{SL}_2(\mathbb{C})}$ action on $\mathbb R^2 \setminus \{ \mathbf{0} \}$, albeit our uniformly nice bounds are obtained only outside of a set of null Lebesgue measure.
