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

DOI:
https://doi.org/10.1090/S0002-9939-2011-11105-5

Published electronically:
November 28, 2011

MathSciNet review:
2910746

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that a large class of Cantor-like sets of , contains uncountably many badly approximable numbers, respectively badly approximable vectors, when . An analogous result is also proved for subsets of arising in the study of geodesic flows corresponding to -dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in . 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:
https://doi.org/10.1090/S0002-9939-2011-11105-5

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.