Size distribution and the Hausdorff-Besicovitch dimension of two-scale Cantor dusts

Simple fractal sets (for example, Cantor dust) can be characterized by a distribution function of sizes of the set’s “building blocks.” This characterization can be useful in problems of fractal growth and coarsening. We test it on a simple example of a two-scale deterministic Cantor dust. In the limit of [Formula Presented] (where m is the number of iterations in the fractal generating algorithm), the discrete binomial distribution of sizes of this set can be approximated by a continuous distribution. This continuous distribution gives an accurate estimate for the Hausdorff-Besicovitch dimension. An algorithm is suggested for generating a random two-scale Cantor dust with a tunable fractal dimension.