Beresnevich–Haynes–Velani inhomogeneous Gallagher conjecture

Let $\psi:\mathbb N\to[0,\infty)$ be an approximating function, let $\bftheta\in\mathbb R^d$, and let $\WW^d_{\psi,\bftheta}\subset\mathbb R^d$ denote the set of points $\xx$ for which $\|qx_1-\theta_1\|\cdots\|qx_d-\theta_d\|<\psi(q)$ for infinitely many $q\in\mathbb N$. Write $\I^d=[0,1]^d$, and normalize $d$-dimensional Hausdorff measure so that $\HH^d(\I^d)=1$. Beresnevich–Haynes–Velani conjecture. For every approximating function $\psi$,

$$\HH^d(\WW^d_{\psi,\bftheta}\cap\I^d)=1\quad\text{if}\quad\sum_{q=1}^{\infty}\psi(q)\log^{d-1}(q)=\infty.$$

The convergence counterpart follows from the first Borel–Cantelli lemma, while the divergence statement is the inhomogeneous analogue of Gallagher's zero–full law and was presented as an open problem.