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Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory

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JournalInternational Mathematics Research Notices
DateAccepted/In press - 23 Dec 2015
DateE-pub ahead of print - 14 Jun 2016
DatePublished (current) - 1 May 2017
Issue number10
Volume2017
Number of pages24
Pages (from-to)2885-2908
Early online date14/06/16
Original languageEnglish

Abstract

In this article, we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold M ⊂ R^n is of dimension strictly greater than (n + 1)/2 and satisfies a natural non-degeneracy condition, then M is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.

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© 2016, The Author(s).

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