Affine mappings of translation surfaces: shrinking targets and Diophantine properties

Let $(X,\omega )$ be a translation surface whose Veech group $\mathrm{\Gamma }$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,\omega )$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $S{L}_2(\mathbb{R})$-action on a fiber bundle $Y$ whose base is $S{L}_2(\mathbb{R})\mathrm{/}\mathrm{\Gamma }$ and whose fiber is $X$. We observe that this bundle embeds as an $S{L}_2(\mathbb{R})$-orbit closure in the moduli space of once marked translation surfaces, and hence we may invoke the spectral gap results of Avila and Gouëzel and a quantitative mean ergodic theorem for the $S{L}_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y$.