  The Jarn\'ik-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
$h>0, \tau\geq 1$, and for any $j\geq 1$ define the integer sequence
$q_{j+1}=q_j^h$. We prove the Hausdorff dimension of the set
$$\Lambda^\bftheta_d(\tau)=\left\{\xx\in[0, 1)^d:
\|q_jx_i-\theta_i\|<q_j^{-\tau} \ \text{for all } j\geq 1,
i=1,2,\cdots,d\right\},$$ where $\left\|\star\right\|$ denotes the distance to
the nearest integer and $\bftheta\in [0, 1)^d$ is fixed. We also give some
heuristics for the Hausdorff dimension of the corresponding multiplicative set
$$\MM_d^\bftheta(\tau)=\left\{\xx\in[0, 1)^d:\prod_{i=1}^d
\|q_jx_i-\theta_i\|<q_j^{-\tau} \ \text{for all } j\geq 1\right\}.$$
