Intrinsic Diophantine approximation on manifolds: General theory

Lior Fishman, Dmitry Kleinbock, Keith Merrill, David Simmons

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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.
Original languageEnglish
Pages (from-to)577-599
Number of pages23
JournalTransactions of the American Mathematical Society
Early online date19 Jul 2017
Publication statusPublished - Jan 2018

Bibliographical note

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


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