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Abstract

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 $\varepsilon $-connected set $E_{\varepsilon }$, we consider a new method based upon coverings of $E_{\varepsilon }$ with crosses of diameter $2{\varepsilon }$. 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.