Diophantine approximation and badly approximable sets

Simon Kristensen, Rebecca Thorn, Sanju Velani
2006 Advances in Mathematics  
Let (X, d) be a metric space and ( , d) a compact subspace of X which supports a nonatomic finite measure m. We consider 'natural' classes of badly approximable subsets of . Loosely speaking, these consist of points in 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
more » ... we prove that the badly approximable subsets of have full Hausdorff dimension. Applications of our general framework include 133 those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).
doi:10.1016/j.aim.2005.04.005 fatcat:sqxcszbpprfvdinm77mbxukr5m