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 $$W(\mathcal P)=\{\textbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_2})^n(\textbf{x})\in P_n\text{ infinitely often}\}.$$ 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 $\mathrm{dim_H} W(\mathcal P)$, where $\mathrm{dim_H}$ 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 $\mathrm{dim_H} W(\mathcal P)$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{dim_H} W(\mathcal P)$.