Simon Kristensen

Diophantine approximation and badly approximable sets

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal article

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

In: Advances in Mathematics, Vol. 203, No. 1, 2006, p. 132-169.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal article

Harvard

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

APA

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

CBE

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

MLA

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

Vancouver

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

Author

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

Bibtex

@article{869e43403d9211dbbee902004c4f4f50,
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",
year = "2006",
language = "English",
volume = "203",
pages = "132--169",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press",
number = "1",

}

RIS

TY - JOUR

T1 - Diophantine approximation and badly approximable sets

AU - Kristensen,S.

AU - Thorn,R.

AU - Velani,S.

PY - 2006

Y1 - 2006

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).

M3 - Journal article

VL - 203

SP - 132

EP - 169

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -