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Classical metric Diophantine approximation revisited: the KhintchineGroshev theorem
Title  Classical metric Diophantine approximation revisited: the KhintchineGroshev 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  6986 
Abstract  Under the assumption that the approximating function $\psi$ is monotonic, the classical KhintchineGroshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $\psi$approximable matrices in $\mathbb{R}^{mn}$. The famous DuffinSchaeffer 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 KhintchineGroshev theorem. This settles a multidimensional analogue of Catlin's Conjecture.

URL  http://arxiv.org/abs/0811.0809 
Eprint number  arXiv:0811.0809

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Edited 20 Jun 2011  18:04 by vb8
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