We study a shrinking target problem on square-tiled surfaces. We show that the action of a subgroup of the Veech group of a regular square-tiled surface exhibits Diophantine properties. This generalizes the work of Finkelshtein, who studied a similar problem on the flat torus [Fin16].

The author thanks Jayadev Athreya for proposing this project and providing guidance, and to the anonymous referee for many helpful comments. Additionally, the author thanks Alexis Drouot, Dami Lee, Farbod Shokrieh, and Bobby Wilson for helpful discussions, and Chris Judge for helpful comments regarding Theorem 3.1. Additionally, the author thanks Lior Silberman for identifying an error in an earlier version of this work, as well as for a series of informative discussions on the representation theory of groups of operators on singular spaces.