Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem

TL;DR

This paper resolves the remaining case of the Khintchine-Groshev theorem in metric Diophantine approximation, removing all unnecessary conditions and settling a multi-dimensional analogue of Catlin's Conjecture.

Contribution

It completes the proof of the Khintchine-Groshev theorem for the case n=2, removing the monotonicity restriction and settling a long-standing conjecture.

Findings

The theorem holds without monotonicity assumption when n=2.

It confirms the multi-dimensional analogue of Catlin's Conjecture.

The result unifies the understanding of Diophantine approximation in multiple dimensions.

Abstract

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 the set of $\psi$-approximable matrices in $\R^{mn}$. 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 $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (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 multi-dimensional analogue of Catlin's Conjecture.