Geodesic
Rational points near manifolds and metric Diophantine approximation
Victor Beresnevich1
1 Department of Mathematics, University of York, Heslington, York YO10 5DD, England
Annals of mathematics, Tome 175 (2012) no. 1, pp. 187-235.

Voir la notice de l'article provenant de la source Annals of Mathematics website

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of $\mathbb{R}^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds — a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).
DOI : 10.4007/annals.2012.175.1.5

Victor Beresnevich 1

1 Department of Mathematics, University of York, Heslington, York YO10 5DD, England 
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Victor Beresnevich. Rational points near manifolds and metric Diophantine approximation. Annals of mathematics, Tome 175 (2012) no. 1, pp. 187-235. doi : 10.4007/annals.2012.175.1.5. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.1.5/

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