The generalized Baker–Schmidt conjecture for simultaneous approximation on nondegenerate manifolds

Let $\psi:\mathbb{N}\to\mathbb{R}^+$ be a decreasing approximation function, and define

$$\mathscr{S}_n(\psi):=\{\mathbf{x}\in\mathbb{R}^n:\text{ there exist infinitely many }q\in\mathbb{Z}\text{ such that }\max_{1\le i\le n}\|qx_i\|<\psi(\lvert q\rvert)\}.$$

Here $\|\cdot\|$ denotes distance to the nearest integer. Let $k$ be the codimension of an $m$-dimensional submanifold $\mathcal{M}\subset\mathbb{R}^n$, so $k=n-m$, and suppose that $\mathcal{M}$ is non-degenerate everywhere except possibly on a set of zero Hausdorff $s$-measure, with $s>\frac{k}{k+1}m$. The generalized Baker–Schmidt conjecture for simultaneous approximation.

$$\mathcal{H}^s(\mathscr{S}_n(\psi)\cap\mathcal{M})= \begin{cases} 0,&\displaystyle\sum_{q=1}^{\infty}\left(\frac{\psi(q)}{q}\right)^{s+k}q^n<\infty,\\ \mathcal{H}^s(\mathcal{M}),&\displaystyle\sum_{q=1}^{\infty}\left(\frac{\psi(q)}{q}\right)^{s+k}q^n=\infty. \end{cases}$$

This is one of the generalized Baker–Schmidt problems in metric Diophantine approximation; the supplied text presents it as a central conjecture and gives no resolution.