Shrinking parallelepiped targets in beta-dynamical systems

Abstract

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 $\mathrm{dim_H} W(\mathcal P)$.