An exponentially shrinking problem

The Jarník-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for some limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let τ>0 and {qj}j≥1 be a sequence of integers. We calculate the Hausdorff dimension of the set Λdθ(τ)={x∈[0,1)d:‖qjxi−θi‖