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