Shrinking parallelepiped targets in beta-dynamical systems

For $\beta >1$ let $T_{\beta }$ be the $\beta$-transformation on $[0,1)$. Let ${\beta }_1,\dots ,{\beta }_d>1$ and let $\mathcal{P}=\{P_n{\}}_{n\ge 1}$ be a sequence of parallelepipeds in $[0,1{)}^d$. Define $$W(\mathcal{P})=\{\mathbf{\text{x}}\in [0,1{)}^d:(T_{{\beta }_1}\times \cdots \times T_{{\beta }_2}{)}^n(\mathbf{\text{x}})\in P_n\text{ infinitely often}\}\mathrm{.}$$ When each $P_n$ is a hyperrectangle with sides parallel to the axes, the ‘rectangle to rectangle’ mass transference principle by Wang and Wu [Math. Ann. 381 (2021)] is usually employed to derive the lower bound for $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$, where $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}$ denotes the Hausdorff dimension. However, in the case where $P_n$ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$.