The general inhomogeneous Khintchine–Groshev conjecture for critical dimensions

Let $n,m$ be positive integers with $nm=2$, and let $\Psi=(B_q)_{q\in\mathbb{N}}$ be any sequence of balls in $\mathbb{R}^m/\mathbb{Z}^m$. Let $\mathcal A_{n,m}(\Psi)$ denote the corresponding set of points satisfying the associated inhomogeneous approximation condition, equipped with Lebesgue measure $|\cdot|$. The general inhomogeneous Khintchine–Groshev conjecture. If

$$\sum_{q=1}^\infty q^{n-1}|B_q|=\infty,$$

then

$$|\mathcal A_{n,m}(\Psi)|=1.$$

This strengthens the preceding conjecture by allowing the approximation balls to vary in position rather than requiring them to be concentric. The source states that the preceding conjecture would follow from this one; both remain open in the critical cases.