The Upper Densities of Perturbed Cantor Sets

The exact computation of upper densities and Hausdorff centered measures is a fundamental yet difficult problem in fractal geometry; up to now, such exact results have been obtained only for a few classes of self-similar sets and for symmetric perfect sets. In this paper, both quantities are determined exactly for the perturbed Cantor set, a class of non-self-similar Moran sets on the unit interval determined by two sequences of contracting ratios. This construction contains the symmetric perfect set and the non-symmetric Cantor set as special cases. It is proved that, for almost every point of the set with respect to the normalized Hausdorff measure, the upper density equals the reciprocal of an explicit constant depending only on the contracting ratios. As a consequence, the Hausdorff centered measure of the set is given by an explicit constant of the same type.