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

L. Singhal¹

¹ School of Mathematics Tata Institute of Fundamental Research Mumbai 400 005, India

Acta Arithmetica, Tome 177 (2017) no. 1, pp. 53-73.

Résumé

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.

Détail

DOI : 10.4064/aa8370-6-2016

Keywords: upper lower bounds various diophantine exponents associated standard linear actions mathrm mathcal punctured complex plane mathbb setminus mathbf where number field whose ring integers mathcal discrete complex number within unit distance element mathcal results similar those laurent nogueira nbsp mathrm mathbb action mathbb setminus mathbf albeit uniformly nice bounds obtained only outside set null lebesgue measure