Hausdorff Dimension in Exponential Time

In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resource-bounded dimension we show that the class of pm-complete sets for E has dimension 1 in E. Moreover, we show that there are p-m-lower spans in E of dimension H(β) for any rational β between 0 and 1, where H(β) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz’s concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions.