On the average box dimensions of graphs of typical continuous functions

Let X be a bounded subset of R d 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 dim B (graph(f)) and dimB(graph(f)), of the graph graph(f) of a function f. Cu(X) are defined by dim B (graph(f)) = lim inf .0 logN (graph(f)) -log , dimB(graph(f)) = lim sup .0 logN (graph(f)) -log , where N (graph(f)) denotes the number of -mesh cubes that intersect graph(f). Hyde et al. have recently proved that the box counting function (*) logN (graph(f)) -log of the graph of a typical function f. Cu(X) diverges in the worst possible way as . 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 . 0, but it is so irregular that it remains spectacularly divergent as . 0 even after being " averaged" or " smoothened out" using exceptionally powerful averaging methods including all higher order Holder and Ces` aro 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 Holder and Ces` aro 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 Holder and Ces` aro averages of the box counting function (*) are as big as possible, namely, equal to the box dimension of X plus 1.