Integral formula for the fractal dimension in the plane (Q2746831)

Let \(z: [a, b] \to {\mathbb R}\) be a continuous, non-constant function and let \(E \subset {\mathbb R}^2\) be the graph set of \(z\). \textit{B. Dubuc} and \textit{C. Tricot} [C. R. Acad. Sci., Paris, Sér. I 306, No. 13, 531-533 (1988; Zbl 0654.26011)] proved that the box dimension of \(E\) is given by the following formula NEWLINE\[NEWLINE \Delta(E) = \limsup_{\varepsilon \to 0} \Big(2 - {{\log \int_a^b \text{osc}(z, \varepsilon, t) dt} \over {\log \varepsilon}}\Big),NEWLINE\]NEWLINE where \(\text{osc}(z, \varepsilon, t)=\sup\{z(t')-z(t''): |t-t'|\leq \varepsilon, |t-t''|\leq \varepsilon \}\). In the paper under review, the author proves that for any compact set \(E \subset {\mathbb R}^2\), NEWLINE\[NEWLINE \Delta(E) = \limsup_{\varepsilon \to 0} \Big(2 - {{\log \big(\int_{\mathbb R} f_\varepsilon(s) ds + \int_{\mathbb R} g_\varepsilon(s) ds\big) } \over {\log \varepsilon}} \Big),NEWLINE\]NEWLINE where \(f_\varepsilon(s)\) is a natural extension of \(\text{osc}(z, \varepsilon, t)\) to any compact set in \({\mathbb R}^2\). Please see the paper for the precise definitions of \(f_\varepsilon(s)\) and \(g_\varepsilon(s)\).