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Singularity of self-similar measures with respect to Hausdorff measures. (English) Zbl 0899.28002
This paper is motivated by the works of A. S. Besicovitch [Math. Ann. 110, 321-330 (1934; Zbl 0009.39503)] and H. G. Eggleston [Q. J. Math., Oxf. Ser. 20, 31-36 (1949; Zbl 0031.20801)] on the subsets of points of the unit interval with given frequencies in the figures of their base-\(p\) expansion. The authors extend the analysis to self-similar sets in \(\mathbb R^N\) and introduce the so-called normal Besicovitch sets by replacing the frequencies of figures with the frequencies of the generating similitudes. Their objective is to analyze the relationships among the normal Besicovitch sets, self-similar measures, and Hausdorff measures. The authors characterize those \(\phi\)-Hausdorff measures with respect to which a self-similar measure is singular or absolutely continuous. They also calculate the Hausdorff dimensions of the normal Besicovitch sets which generalize the results of Besicovitch and Eggleston, and show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the law of the iterated logarithm.

MSC:
28A78 Hausdorff and packing measures
28A80 Fractals
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