Settle weighted bad-approximation problems for affine subspaces in dimensions greater than two

Determine sufficient conditions on an affine map F from a ball in a Euclidean space to \(\mathbb{R}^n\), when \(n>2\), that ensure both full Hausdorff dimension of the intersection of the relevant inverse images \(F^{-1}(\mathbf{Bad}(w))\) over prescribed collections of weights and maps, and the winning property of \(F^{-1}(\mathbf{Bad}(w))\) for every weight \(w\).

Background

The introduction recalls two problems concerning weighted badly approximable vectors on parametrized sets. The first asks for sufficient, or even necessary, conditions guaranteeing that intersections of inverse images of weighted badly approximable sets have full Hausdorff dimension. The second, identified there as Conjecture D of Badziahin–Velani, asks for sufficient conditions on a map $F$ ensuring that $F^{-1}(\mathbf{Bad}(w))$ is winning for every weight.

For affine subspaces in dimensions greater than two, the authors state that neither problem had been settled at the time of writing. The paper establishes a hyperplane absolute winning result under an almost optimal weighted Diophantine condition on the affine subspace, thereby resolving a substantial conditional case rather than the unrestricted affine problem.