Pseudofractals and the box counting algorithm

We show that, for sets with the Hausdorff-Besicovitch dimension equal to zero, the box counting (BC) algorithm commonly used to calculate the Renyi exponents (dq) can exhibit perfect scaling suggesting non-zero dq. The properties of these pathological sets (pseudofractals) are investigated. Numerical, as well as analytical, estimates for dq are obtained. A simple indicator is given to distinguish pseudofractals and fractals in practical applications of the BC method. Histograms made of pseudofractal sets are shown to have Pareto tails. (author)