Hausdorff dimension in exponential time

Abstract

In this paper we investigate effective versions of Hausdo$ dimension which have been recently introduced byLutz. 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 immuniv. By a new general invariance theorem f o r resource-bounded dimension we show that the class of pm-complete sets for E has dimension 1 in E. Moreovel; we show that there are p-m-lower spans in E of dimension X ( P ) for any rational /3 between 0 and 1, where X ( P ) 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.In 1919, HAUSDORFF [8] introduced a theory of dimension which generalizes Lebesgue measure and, in particular, ramifies the structure of measure zero sets. Hausdog dimension, as it was called afterwards, found applications in many areas of mathematics such as number theory or dynamical systems (see for instance [7]). Probably bestknown is the role Hausdorff dimension plays in fractal geometry, where it turned out to be a suitable concept for distinguishing fractal sets from sets of a rather "smooth" geometrical nature.In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4]), a close connection between Hausdorff dimension and Kolmogorov complexity was established [6,16, 17,18,20, 211, and the notion of effective dimension was introduced [ 141. Recently, LUTZ [ 131 has extended the theory of effective dimension to complexity theory by introducing resource-bounded dimension. Like

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