Number one in the UK and seventh in the world in the Times Higher Education rankings of universities under 50 years old


Athena SWAN Bronze Award

Supporter of LMS Good Practice Award

Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem

Research group:
TitleClassical metric Diophantine approximation revisited: the Khintchine-Groshev theorem
Publication TypeJournal Article
Year of Publication2010
AuthorsBeresnevich V, Velani S
Refereed DesignationRefereed
JournalInt. Math. Res. Not.
Volume2010, no.1
AbstractUnder 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.
E-print numberarXiv:0811.0809

Edited 20 Jun 2011 - 19:04 by vb8

Back to the Top