Diophantine approximation and badly approximable sets

Simon Kristensen; Rebecca Thorn; S. Velani

doi:10.1016/j.aim.2005.04.005

Diophantine approximation and badly approximable sets

Simon Kristensen, Rebecca Thorn, S. Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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

Original language	English
Pages (from-to)	132-169
Number of pages	37
Journal	Advances in Mathematics
Volume	203
Issue number	1
DOIs	https://doi.org/10.1016/j.aim.2005.04.005
Publication status	Published - 20 Jun 2006

Keywords

diophantine approximation
Hausdorff dimension
badly approximable elements
dynamical systems
ERGODIC-THEORY
NUMBERS

Access to Document

10.1016/j.aim.2005.04.005

http://dx.doi.org/10.1016/j.aim.2005.04.005

Cite this

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title = "Diophantine approximation and badly approximable sets",

abstract = "Let (X,d) be a metric space and (O,d) a compact subspace of X which supports a non-atomic finite measure m. We consider {\textquoteleft}natural{\textquoteright} classes of badly approximable subsets of O. Loosely speaking, these consist of points in O which {\textquoteleft}stay clear{\textquoteright} of some given set of points in X. The classical set Bad of {\textquoteleft}badly approximable{\textquoteright} 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 O 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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T1 - Diophantine approximation and badly approximable sets

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AU - Thorn, Rebecca

AU - Velani, S.

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

KW - diophantine approximation

KW - Hausdorff dimension

KW - badly approximable elements

KW - dynamical systems

KW - ERGODIC-THEORY

KW - NUMBERS

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DO - 10.1016/j.aim.2005.04.005

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