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Diophantine approximation and badly approximable sets

Kristensen, S., Thorn, R. and Velani, S. (2006) Diophantine approximation and badly approximable sets. Advances in Mathematics, 203 (1). pp. 132-169. ISSN 0001-8708

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Abstract

Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).

Item Type: Article
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 13 Aug 2009 14:29
Last Modified: 13 Aug 2009 14:29
Published Version: http://dx.doi.org/10.1016/j.aim.2005.04.005
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.aim.2005.04.005
URI: http://eprints.whiterose.ac.uk/id/eprint/5484

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