Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup

We develop the Mass Transference Principle for rectangles of Wang & Wu (Math. Ann. 2021) to incorporate the unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring $\limsup$ sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for $τ>0$, let $S(τ)$ denote the set of $(x_1,x_2)\in \mathbb{R}^2$ simultaneously satisfying the inequalities $\|q x_1 \| \, < \, q^{-τ} $ and $ \|q x_2 \| \, < \, e^{-q}$ for infinitely many $q \in \mathbb{N}$. Then, the unbounded’ Mass Transference Principle enables us to show that ${\dim }_{\mathrm{H}}S(\tau )=\min \{1,3\mathrm{/}(1+\tau )\}$.