Algebraic structures featuring graph dimensions, H\"older regularity, and fractional differentiability

TL;DR

This paper explores the algebraic structure of irregular continuous functions, demonstrating the strong algebrability of functions with specific fractal dimensions, H"older regularity, and fractional differentiability properties.

Contribution

It establishes the strong $rak{c}$-algebrability of functions with prescribed fractal dimensions and fractional differentiability, addressing open questions and extending previous results.

Findings

Functions with graph dimension s form a strongly $rak{c}$-algebrable set.

H"older functions with constant pointwise exponent are $rak{c}$-lineable but not algebrable.

Families of functions with H"older exponent outside a zero Hausdorff dimension set are strongly $rak{c}$-algebrable.

Abstract

We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, H\"older 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 H\"older spaces: although the pointwise H\"older exponent of a generic function in $C^{\alpha}[0, 1]$ 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 H\"older exponent $\alpha$ outside a set of Hausdorff dimension zero. Finally, as a consequence of the relation between strongly monoH\"older functions and fractional differentiability, we analyze the strong $\mathfrak{c}$-algebrability of nowhere (Riemann-Liouville) fractional differentiable functions.