An exponentially shrinking problem

Abstract

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 $$\MM_d^\bftheta(\tau)=\left\{\xx\in[0, 1)^d:\prod_{i=1}^d \|q_jx_i-\theta_i\|<q_j^{-\tau} \ \text{for all } j\geq 1\right\}.$$