Rational points near manifolds and metric Diophantine approximation

By Victor Beresnevich. 2009

113 citations recorded by the source corpus.

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate 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 non-degenerate manifolds in dimensions n>2 and show that they are ubiquitous (that is uniformly distributed).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d8b8177e6d6b12fbb17d415c26473e275bb540f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Badly approximable points on manifolds (2013, background): “In the theory of Diophantine approximation on manifolds, see for instance, [7–9,31,33], there are already well established classes of manifolds of interest.”
Metric Diophantine Approximation : aspects of recent work (2016, background): “The divergence case of the Hausdorff Dream Theorem is true for analytic non-degenerate sub-manifolds of Rn [10].”