Mathematics – Number Theory
Scientific paper
2004-05-24
Mathematics
Number Theory
Final version, to appear in Adv. Math
Scientific paper
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).
Kristensen Simon
Thorn Rebecca
Velani Sanju
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