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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Badly approximable numbers and vectors in Cantor-like sets


Authors: S. G. Dani and Hemangi Shah
Journal: Proc. Amer. Math. Soc. 140 (2012), 2575-2587
MSC (2010): Primary 11J25, 37D40, 37C35
Published electronically: November 28, 2011
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Abstract: We show that a large class of Cantor-like sets of $ {{\mathbb{R}}}^{d}, d \geq 1$, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when $ d\geq 2$. An analogous result is also proved for subsets of $ {\mathbb{R}}^d$ arising in the study of geodesic flows corresponding to $ (d+1)$-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in $ {\mathbb{R}}$. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.


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Additional Information

S. G. Dani
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Email: dani@math.tifr.res.in

Hemangi Shah
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Email: hemangi@math.iisc.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11105-5
PII: S 0002-9939(2011)11105-5
Received by editor(s): February 7, 2011
Received by editor(s) in revised form: March 3, 2011
Published electronically: November 28, 2011
Additional Notes: The second author thanks the Tata Institute of Fundamental Research, Mumbai, and the National Board for Higher Mathematics for support through Research Fellowships while this work was being done.
The authors thank the referee for helpful suggestions.
Communicated by: Bryna Kra
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.