Approximation of points in the plane by generic lattice orbits

Abstract

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma <{\mathrm{SL}}_2(\mathbb{R})$ acting linearly on ${\mathbb{R}}^2$. Our method gives bounds that are uniform for almost all orbits.