A method for evaluating the fractal dimension in the plane, using coverings with crosses (Q2773383)

Box-counting or Minkowski dimension is the most convenient fractal dimensional index in applications. Nice estimates, however, are not easily accessible from logarithmic plots using classical definitions. A new method to compute the box-counting dimension of plane compact sets is proposed in this paper. Motivated by a definition of box dimension for graphs of continuous functions, based on evaluating area of centered crosses of small diameter covering the graph [\textit{C. Tricot}, ``Curves and fractal dimension'' (1995; Zbl 0847.28004)], the author derives a version valid for arbitrary compact sets. An equivalent integral formula, generalizing the integral formula for the box dimension of graphs, is also obtained. Numerical evidence of the efficiency of the method was shown elsewhere for the case of graphs [\textit{B. Dubuc, J. F. Quiniou, C. Roques-Carmes, C. Tricot} and \textit{S. W. Zucker}, Phys. Rev. A 39, 1500-1512 (1989; MR 90a:58101)].