Shrinking targets on square-tiled surfaces (Q6555315)

In 1995, Hill and Velani coined the term ''shrinking target'' in their fundamental work on the subject [\textit{E. Gutkin} and \textit{C. Judge}, Math. Res. Lett. 3, No. 3, 391--403 (1996; Zbl 0865.30060)]. In 2016, \textit{V. Finkelshtein} [``Diophantine properties of groups of toral automorphisms'', Preprint, \url{arXiv:1607.06019}] studied a shrinking target problem on the square torus. The torus is an example of a translation surface and \({\mathrm{SL}}_2({\mathbb{Z}})\) is its Veech group. Moreover, \({\mathrm{SL}}_2({\mathbb{Z}})\) is a lattice subgroup of \({\mathrm{SL}}_2({\mathbb{{\mathbb{R}}}})\), so the torus is an example of a lattice surface. Finkelshtein showed that the action of \({\mathrm{SL}}_2({\mathbb{Z}})\)on the torus exhibits certain Diophantine estimates. Finkelshtein's proof relies on a fundamental connection between the dynamics of the Veech group action and the Laplacian on the torus.\N\NHere is the main result of the paper under review. Let \((X,\omega)\) be a regular square-tiled surface, and let \(\Gamma\) be a subgroup of the Veech group \({\mathrm{SL}}(X,\omega)\) with critical exponent \(\delta_\Gamma>0\). For any \(y\in X\) for Lebesgue a.e. \(x\in X\), the set\N\[\N\{ g\in \Gamma\; : \; |gx-y|<\Vert g\Vert^{-\alpha}\}\N\]\Nis \\\N(1) finite for every \(\alpha>\delta_\Gamma\), \\\N(2) infinite for every \(\alpha<\delta_\Gamma\), \\\Nwhere \(\Vert g \Vert\) is the operator norm of \(g\) (as a linear transformation on \({\mathbb{R}}^2\)).