  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.
