Measures and the Law of the Iterated Logarithm

Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this question in several situations (self-similar measures, quasi-Bernoulli measures). More precisely we obtain fine comparisons between the mesure m and generalized Hausdorff type (or packing type) measures. The Law of the Iterated Logarithm or estimations of the L^q-spectrum in a neighborhood of q=1 are the tools to obtain such results.