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Rational points near manifolds and metric Diophantine approximation

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Publication details

JournalAnnals of Mathematics
DateE-pub ahead of print - 2 Apr 2009
DatePublished (current) - Jan 2012
Issue number1
Volume175
Number of pages49
Pages (from-to)187-235
Early online date2/04/09
Original languageEnglish

Abstract

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).

    Research areas

  • Number Theory, simultaneous Diophantine approximation on manifolds, , metric theory, Khintchine theorem, Jarnik theorem, Hausdorff dimension, ubiquitous systems, rational points near manifolds

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