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Diophantine approximation and badly approximable sets

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  • S. Kristensen
  • R. Thorn, Queen Mary Universty of London, United Kingdom
  • S. Velani, University of York, United Kingdom
  • Department of Mathematical Sciences
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).
Original languageEnglish
JournalAdvances in Mathematics
Pages (from-to)132-169
Number of pages38
Publication statusPublished - 2006

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