On the average box dimensions of graphs of typical continuous functions. (English) Zbl 1413.28006
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 \(n\in\mathbb{N}\cup \{0\}\).
Reviewer: Hans Weber (Udine)
Keywords:
box dimension; continuous function; Hölder mean; Cesáro mean; Riesz-Hardy mean; Baire categoryReferences:
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