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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Intrinsic Diophantine approximation on manifolds: General theory


Authors: Lior Fishman, Dmitry Kleinbock, Keith Merrill and David Simmons
Journal: Trans. Amer. Math. Soc. 370 (2018), 577-599
MSC (2010): Primary 11J13, 11J83; Secondary 37A17
DOI: https://doi.org/10.1090/tran/6971
Published electronically: July 19, 2017
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Abstract: 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.


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Additional Information

Lior Fishman
Affiliation: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, Texas 76203-5017
Email: lior.fishman@unt.edu

Dmitry Kleinbock
Affiliation: Department of Mathematics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454-9110
Email: kleinboc@brandeis.edu

Keith Merrill
Affiliation: Department of Mathematics, Brandeis University, 415 South Street, Waltham, Massachusetts 02454-9110
Email: merrill2@brandeis.edu

David Simmons
Affiliation: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
Email: David.Simmons@york.ac.uk

DOI: https://doi.org/10.1090/tran/6971
Received by editor(s): September 29, 2015
Received by editor(s) in revised form: April 13, 2016
Published electronically: July 19, 2017
Additional Notes: The first-named author was supported in part by the Simons Foundation grant #245708. The second-named author was supported in part by the NSF grant DMS-1101320. The fourth-named author was supported in part by the EPSRC Programme Grant EP/J018260/1.
Article copyright: © Copyright 2017 American Mathematical Society

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