Shrinking parallelepiped targets for $\beta $-dynamical systems

Abstract For $ \beta>1 $ , let $ T_\beta $ be the $\beta $ -transformation on $ [0,1) $ . Let $ \beta _1,\ldots ,\beta _d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $ . Define $$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$ When each $ P_n $ is a hyperrectangle with sides parallel to the axes, the ‘rectangle to rectangle’ mass transference principle by Wang and Wu [Mass transference principle from rectangles to rectangles in Diophantine approximation. Math. Ann. 381 (2021) 243–317] is usually employed to derive the lower bound for $\dim _{\mathrm {H}} W(\mathcal P)$ , where $\dim _{\mathrm {H}}$ 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 $\dim _{\mathrm {H}} W(\mathcal P)$ . We also provide several examples to illustrate how the rotations of hyperrectangles affect $\dim _{\mathrm {H}} W(\mathcal P)$ .