A localized Jarnik-Besicovitch theorem
Résumé
Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x is an element of R: delta(x) = delta}, where delta >= I and delta(x) 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 is an element of R: delta(x) = 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. (C) 2010 Elsevier Inc. All rights reserved.