On the average box dimensions of graphs of typical continuous functions
For a bounded set \(E\subseteq \mathbb{R}^m\) and \(\delta >0\) let \(N_\delta(E)\) be the number of \(\delta\)-mesh cubes that intersect \(E\) and \(\Delta(E,\delta)=\frac{\log N_\delta(E)}{-\log\delta}\). The lower and upper box dimensions are defined by \(\underline{\dim}_B (E)=\liminf_{\delta\searrow 0}\Delta(E,\delta)\) and \(\overline{\dim}_B (E)=\limsup_{\delta\searrow 0}\Delta(E,\delta)\). Let \(X\) be a bounded subset of \(\mathbb{R}^d\) and \(C_u(X)\) the Banach space of all uniformly continuous real-valued functions on \(X\) equipped with the uniform norm \(\|\;\|_\infty\). It is known that \(\underline{\dim}_B(X)\leq \underline{\dim}_B(\text{graph}(f))\leq \overline{\dim}_B(\text{graph}(f))\leq \overline{\dim}_B(X)+1\) for \(f\in C_u(X)\) and \(\underline{\dim}_B(\text{graph}(f))=\underline{\dim}_B(X)\) for a typical function \(f\in C_u(X)\); moreover, if \(X\) has no isolated points, then \(\overline{\dim}_B(\text{graph}(f))=\overline{\dim}_B(X)+1\) for a typical function \(f\in C_u(X)\). Here the authors say that a typical function \(f\in C_u(X)\) has the property \(P\) if \(\{f\in C_u(X): f \text{ has property } P\}\) is co-meager. The aim of the paper is to show: ``not only is the box counting function \(\Lambda_f(\delta):=\Delta(\text{graph}(f),\delta)\) divergent as \(\delta\searrow 0\), but it is so irregular that it remains spectacularly divergent as \(\delta\searrow 0\) even after being ``averaged''. The authors illustrate their very general results by the following particular case: Define \(\Lambda_f^n(t)\) inductively by \(\Lambda_f^0(t)=\Lambda_f(e^{-t})\), \(\Lambda_f^n(t)=\frac{1}{t}\int_1^t\Lambda_f^{n-1}(s)ds\). Then a typical continuous function \(f:[0,1]^d\rightarrow\mathbb{R}\) satisfies \(\liminf_{t\rightarrow\infty}\Lambda_f^n(t)=d\), \(\limsup_{t\rightarrow\infty}\Lambda_f^n(t)=d+1\) for all \(n\in\mathbb{N}\cup \{0\}\).
scientific article; zbMATH DE number 7011031