Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem

Title	Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem
Publication Type	Journal Article
Year of Publication	2010
Authors	Beresnevich V, Velani S
Refereed Designation	Refereed
Journal	Int. Math. Res. Not.
Volume	2010, no.1
Pagination	69-86
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 $\mathbb{R}^{mn}$. 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.
URL	http://arxiv.org/abs/0811.0809
E-print number	arXiv:0811.0809

Department of Mathematics