Hausdorff dimension for prescribed growth rates

Determine the Hausdorff dimension of \(E_Q=\{(x,y)\in\mathbb R^2:\liminf_{m\to\infty}Q_m\min_{1\le q\le Q_m}\|qx\|\,\|qy\|>0\}\) for an arbitrary prescribed growth rate of the strictly increasing sequence \(Q=(Q_m)_{m\ge0}\) tending to infinity.

Background

The set $E_Q$ is a quantitative variant of the counterexample set for uniform Littlewood's conjecture, obtained by sampling the scale parameter along a sequence $Q_m$. The paper determines the dimension in two broad regimes: bounded ratios yield dimension zero, while sufficiently superlinear growth yields full dimension two and, in fact, the hyperplane absolute winning property. The authors explicitly pose the dimension question for general growth rates as a problem.