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 expansions. We extend this analysis to self-similar sets, by replaci ng the frequencies of figures with the frequencies of the generating s imilitudes. We focus on the interplay among such sets, self-similar me asures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-simil ar measures with respect to those measures. We show that the self-simi lar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.