A Localized Jarnik-besicovitch Theorem

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x ∈ R : δx = δ}, where δ ≥ 1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x ∈ R : δx = f(x)}, where f is a continuous function. Our theorem applies to the study of the approximation exponents by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension.