Well and badly approximable sets, and rapid winning

Abstract: The set of $τ$-approximable numbers, $\mathcal W(τ)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad<sup>γ$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $Ψ$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $ψ$, we prove that $\mathcal W(ψ)\cap\Bad<sup>γ$ is strong $Ψ$-rapid winning for a natural gauge $Ψ$ determined by $ψ$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $Ψ$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $Ψ$. In particular, for $ψ(q)=q^{-τ}, τ\ge1,$ we recover the exact Jarník--Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(τ)\cap\Bad<sup>γ\bigr)=\frac{2}{τ+1}.$$