Higher-dimensional inhomogeneous multiplicative Gallagher conjecture

Let $m,d\geq1$ be integers, let $\Psi:\mathbb Z^m\to(0,\infty)$, and fix $\bftheta\in\I^d$. For $X=(\mathbf x^{(1)},\ldots,\mathbf x^{(d)})\in\I^{md}$, define $\WW^{md}_{\Psi,\bftheta}$ by the condition

$$\prod_{i=1}^d\|\mathbf q\cdot\mathbf x^{(i)}-\theta_i\|<\Psi(\mathbf q)$$

for infinitely many $\mathbf q\in\mathbb Z^m\setminus\{\mathbf0\}$. Let $|\mathbf q|$ be the sup norm. Higher-dimensional inhomogeneous multiplicative Gallagher conjecture. If $\Psi(\mathbf q)=\psi(|\mathbf q|)$ for a monotonic function $\psi$, then

$$\HH^{md}(\WW^{md}_{\Psi,\bftheta})=1$$

whenever $\sum_{\mathbf q\in\mathbb Z^m\setminus\{\mathbf0\}}\Psi^m(\mathbf q)\log^{d-1}(|\mathbf q|)$ diverges. Apart from the homogeneous zero–one law, the paper states that no corresponding theory was known, so this is an open higher-dimensional extension.