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**Diophantine approximation and badly approximable sets.** / Kristensen, S.; Thorn, R.; Velani, S.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Kristensen, S, Thorn, R & Velani, S 2006, 'Diophantine approximation and badly approximable sets' *Advances in Mathematics*, vol. 203, no. 1, pp. 132-169.

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

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

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

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

Kristensen, S. ; Thorn, R. ; Velani, S. / **Diophantine approximation and badly approximable sets**. In: Advances in Mathematics. 2006 ; Vol. 203, No. 1. pp. 132-169.

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title = "Diophantine approximation and badly approximable sets",

abstract = "Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega 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 Omega 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).",

author = "S. Kristensen and R. Thorn and S. Velani",

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AU - Velani, S.

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N2 - Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega 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 Omega 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).

AB - Let (X,d) be a metric space and (Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Omega. Loosely speaking, these consist of points in Omega 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 Omega 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).

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JO - Advances in Mathematics

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