Dichotomy laws for the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems

In this paper, we investigate the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well-understood, the Hausdorff measure theory in even one-dimensional settings remains unknown. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.