Sharp arithmetic conditions for full-dimensional reversed intersections

Determine sharp arithmetic conditions on \((\gamma_0,\gamma_1,\ldots,\gamma_r)\) under which, for \(\tau\geq1\), the Hausdorff dimension of \(W^{\gamma_0}(\tau)\cap\bigcap_{j=1}^rBad^{\gamma_j}\) equals \(2/(\tau+1)\).

Background

The proposed reversed problem concerns simultaneous inhomogeneous approximation to $\gamma_0$ and inhomogeneous bad approximability with respect to shifts $\gamma_1,\ldots,\gamma_r$. The target dimension $2/(\tau+1)$ is the Jarník–Besicovitch dimension for the $\tau$-approximable set established in the paper.

The paper explicitly observes that arithmetic relations among the shifts can force the intersection to be empty, so identifying sharp conditions is necessary before an exact dimension formula can hold.