Algebraic structures featuring graph dimensions, Hölder regularity, and fractional differentiability

Abstract

We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, Hölder regularity, and fractional differentiability. Our first main result shows that for every $s\in (1,s]$, the set of continuous functions on $[0,1]$ whose graph has Hausdorff and box dimensions equal to s is strongly $\mathfrak{c}$-algebrable, thereby tackling an open question by Bonilla et al., and complementing recent findings by Liu et. al and Carmona et al. We then extend the analysis to Hölder spaces: although the pointwise Hölder exponent of a generic function in $C^{\hat{I}}\pm [0,1]$ is constant, we prove that the collection of functions realizing this behavior is $\mathfrak{c}$-lineable but cannot form an algebra. Nevertheless, we construct strongly $\mathfrak{c}$-algebrable families of functions that exhibit Hölder exponent $\hat{I}\pm$ outside a set of Hausdorff dimension zero. Finally, as a consequence of the relation between strongly monoHölder functions and fractional differentiability, we analyze the strong $\mathfrak{c}$-algebrability of nowhere (Riemann-Liouville) fractional differentiable functions.