Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Abstract

Let denote the set of ψ-approximable points in . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine–Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin–Schaeffer counterexample shows that the monotonicity assumption on ψ 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.