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Intrinsic Diophantine approximation on manifolds: General theory

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Publication details

JournalTransactions of the American Mathematical Society
DateAccepted/In press - 5 May 2016
DateE-pub ahead of print - 19 Jul 2017
DatePublished (current) - Jan 2018
Number of pages23
Pages (from-to)577-599
Early online date19/07/17
Original languageEnglish


We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on $M$ distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the Simplex Lemma for rationals lying on $M$ which provides new insights on the local distribution of rational points on nondegenerate manifolds.

Bibliographical note

© 2017 American Mathematical Society. This paper is split off from arXiv:1405.7650v2

    Research areas

  • math.NT


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