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

Various methods may be used to define the Minkowski-Bouligand dimension of a compact subset E in the plane. The best known is the box method. After introducing the notion of epsilon-connected set E-epsilon, we consider a new method based upon coverings of E-epsilon with crosses of diameter 2epsilon. To prove that this cross method gives the fractal dimension for all E, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.