The generalized Baker–Schmidt problem for Hausdorff measure

Let $\psi:\mathbb{N}\to\mathbb{R}^+$ be an approximation function, let $\mathcal{M}\subset\mathbb{R}^n$ be an $m$-dimensional submanifold, and let $s>m-1$. Assume that $\mathcal{M}$ is non-degenerate everywhere except possibly on a set of zero Hausdorff $s$-measure. Define

$$A_n(\psi):=\{\mathbf{x}\in\mathbb{R}^n:\exists^\infty\mathbf{q}\in\mathbb{Z}^n\text{ such that }\|\mathbf{q}\cdot\mathbf{x}\|<\psi(|\mathbf{q}|)\}.$$

Generalized Baker–Schmidt problem for Hausdorff measure.

$$\mathcal{H}^s(A_n(\psi)\cap\mathcal{M})=\begin{cases}0&\text{if }\displaystyle\sum_{q=1}^{\infty}\left(\frac{\psi(q)}q\right)^{s+1-m}q^n<\infty,\\ \mathcal{H}^s(\mathcal{M})&\text{if }\displaystyle\sum_{q=1}^{\infty}\left(\frac{\psi(q)}q\right)^{s+1-m}q^n=\infty.\end{cases}$$

The source presents this as a delicate conjectural generalization and notes the necessity of the exceptional-set hypothesis; its status is not resolved there.