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

We show that if a real x ∈ 2 ω is strongly Hausdorff H h -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π 0 1 -classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman's Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman's Theorem.

Abstract. We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every noncomputable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.

It is well known that Martin-Löf randomness can be characterized by a number of equivalent test concepts, based either on effective nullsets (Martin-Löf and Solovay tests) or on prefix-free Kolmogorov complexity (lower and upper entropy). These equivalences are not preserved as regards the partial randomness notions induced by effective Hausdorff measures or partial incompressibility. Tadaki [21] and Calude, Staiger and Terwijn [2] studied several concepts of partial randomness, but for some of them the exact relations remained unclear. In this paper we will show that they form a proper hierarchy of randomness notions, namely for any ρ of the form ρ(x) = 2 −|x|s with s being a rational number satisfying 0 < s < 1, the Martin-Löf ρ-tests are strictly weaker than Solovay ρ-tests which in turn are strictly weaker than strong Martin-Löf ρ-tests. These results also hold for a more general class of ρ introduced as unbounded premeasures.

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