Relation between spectral classes of a self-similar Cantor set

A self-similar Cantor set is completely decomposed as a class of the lower (upper) distribution sets. We give a relationship between the distribution sets in the distribution class and the subsets in a spectral class generated by the lower (upper) local dimensions of a self-similar measure. In particular, we show that each subset of a spectral class is exactly a distribution set having full measure of a self-similar measure related to the distribution set using the strong law of large numbers. This gives essential information of its Hausdorff and packing dimensions. In fact, the spectral class by the lower (upper) local dimensions of every self-similar measure, except for a singular one, is characterized by the lower or upper distribution class. Finally, we compare our results with those of other authors.