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Under the assumption that the approximating function $\psi$ is monotonic, theclassical Khintchine-Groshev theorem provides an elegant probabilisticcriterion for the Lebesgue measure of the set of $\psi$-approximable matricesin $\R^{mn}$. The famous Duffin-Schaeffer counterexample shows that themonotonicity assumption on $\psi$ is absolutely necessary when $m=n=1$. On theother hand, it is known that monotonicity is not necessary when $n > 2$(Sprindzuk) or when $n=1$ and $m>1$ (Gallagher). Surprisingly, when $n=2$ thesituation is unresolved. We deal with this remaining case and thereby removeall unnecessary conditions from the classical Khintchine-Groshev theorem. Thissettles a multi-dimensional analogue of Catlin's Conjecture.

Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem