Diophantine approximation and badly approximable sets

Kristensen, Simon; Thorn, Rebecca; Velani, Sanju

Diophantine approximation and badly approximable sets

Authors

Kristensen, Simon

Thorn, Rebecca

Velani, Sanju

Description

Let (X,d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ωwhich `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 Ωhave 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).
Final version, to appear in Adv. Math