On the average box dimensions of graphs of typical continuous functions.

Let X be a bounded subset of Rd and write Cu(X) for the set of uniformly continuous functions on X equipped with the uniform norm. The lower and upper box dimensions, denoted by dim̲B(graph(f)) and dim¯B(graph(f)), of the graph graph(f) of a function f∈Cu(X) are defined by dim̲B(graph(f))=limδ↘0inflogNδ(graph(f))-logδ,dim¯B(graph(f))=limδ↘0suplogNδ(graph(f))-logδ,where Nδ(graph(f)) denotes the number of δ-mesh cubes that intersect graph(f).Hyde et al. have recently proved that the box counting function (∗)logNδ(graph(f))-logδof the graph of a typical function f∈Cu(X) diverges in the worst possible way as δ↘0. More precisely, Hyde et al. showed that for a typical function f∈Cu(X), the lower box dimension of the graph of f is as small as possible and if X has only finitely many isolated points, then the upper box dimension of the graph of f is as big as possible.In this paper we will prove that the box counting function (*) of the graph of a typical function f∈Cu(X) is spectacularly more irregular than suggested by the result due to Hyde et al. Namely, we show the following surprising result: not only is the box counting function in (*) divergent as δ↘0, but it is so irregular that it remains spectacularly divergent as δ↘0 even after being “averaged" or “smoothened out" using exceptionally powerful averaging methods including all higher order Hölder and Cesàro averages and all higher order Riesz-Hardy logarithmic averages. For example, if the box dimension of X exists, then we show that for a typical function f∈Cu(X), all the higher order lower Hölder and Cesàro averages of the box counting function (*) are as small as possible, namely, equal to the box dimension of X, and if, in addition, X has only finitely many isolated points, then all the higher order upper Hölder and Cesàro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1. [ABSTRACT FROM AUTHOR]