A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\mathrm{lim\hspace{0.17em}sup}$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\mathrm{lim\hspace{0.17em}sup}$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\mathrm{lim\hspace{0.17em}sup}$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.