Singularity of self-similar measures with respect to Hausdorff measures

Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base– p p expansions. We extend this analysis to self–similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self–similar measures, and Hausdorff measures. We give a fine–tuned classification of the Hausdorff measures according to the singularity of the self–similar measures with respect to those measures. We show that the self–similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.