Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Let A(n,m)(psi) denote the set of psi-approximable points in R-mn. Under the assumption that the approximating function psi is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of A(n,m)(psi). The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on psi is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n >= 3 ( Schmidt) or when n = 1 and m >= 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.