2009/03/12 by Barral, Julien, Seuret, Stephane
#11JXX #11K55 #11K60 #28A78 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0903.2215
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 rate 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 rates by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension.