Shrinking targets on square-tiled surfaces

TL;DR

This paper investigates how certain subgroup actions on square-tiled surfaces demonstrate Diophantine properties, extending previous results from flat tori to more complex surfaces.

Contribution

It generalizes the shrinking target problem from flat tori to square-tiled surfaces, revealing Diophantine behavior of subgroup actions.

Findings

Subgroup actions on square-tiled surfaces exhibit Diophantine properties.

Generalization of Finkelshtein's results from tori to square-tiled surfaces.

New insights into dynamical properties of Veech group actions.

Abstract

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.