Shrinking parallelepiped targets in beta-dynamical systems

Abstract

For ${\hat{I}}^2>1$ let $T_{\hat{I}}^2$ be the ${\hat{I}}^2$-transformation on $[0,1)$. Let ${\hat{I}}_1^2,\dots ,{\hat{I}}_d^2>1$ and let $\mathcal{P}=\{P_n{\}}_{n\ge 1}$ be a sequence of parallelepipeds in $[0,1{)}^d$. Define \[W(\mathcal P)=\{\textbf{x}\in[0,1)^d:(T_{β_1}\times\cdots \times T_{β_2})^n(\textbf{x})\in P_n\text{ infinitely often}\}.\] 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 $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$, where $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}$ 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 $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{di}{\mathrm{m}}_{\mathrm{H}}W(\mathcal{P})$.