An exponentially shrinking problem

The Jarník-Besicovitch theorem is a fundamental result in metric number theory which gives the Hausdorff dimension for limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let $h>0,\tau \ge 1$, and for any $j\ge 1$ define the integer sequence $q_{j+1}=q_j^h$. We prove the Hausdorff dimension of the set $${\mathrm{\Lambda }}_d^{\text{bftheta}}(\tau )=\{\text{xx}\in [0,1{)}^d:\mathrm{\parallel }{q}_j{x}_i-{\theta }_i\mathrm{\parallel }<q_j^{-\tau }\text{ for all }j\ge 1,i=1,2,\cdots ,d\},$$ where $\parallel \star \parallel$ denotes the distance to the nearest integer and $\text{bftheta}\in [0,1{)}^d$ is fixed. We also give some heuristics for the Hausdorff dimension of the corresponding multiplicative set $${\text{MM}}_d^{\text{bftheta}}(\tau )=\{\text{xx}\in [0,1{)}^d:\prod _{i=1}^d\mathrm{\parallel }{q}_j{x}_i-{\theta }_i\mathrm{\parallel }<q_j^{-\tau }\text{ for all }j\ge 1\}\mathrm{.}$$