On strong spaceability of continuous functions and fractal dimensions

Abstract

Given $s\in (1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim }_H{G}_f([0,1])=s\}$$ and $${\overline{B}}_s[0,1]=\{f\in C[0,1]:{\overline{\dim }}_B{G}_f([0,1])=s\}.$$ The main goal of this paper is to study the $(\alpha ,\beta )$-lineability/spaceability of the sets $H_s[0,1]$ and ${\overline{B}}_s[0,1]$. As a principal result, we prove that $H_s[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$ and also $(n,n+m)$-lineable for any $m,n\in \mathbb{N}$. This partially answers a question raised by Liu et al. concerning the Hausdorff dimension of graphs of continuous functions. Furthermore, for a cardinal number $\alpha$, we prove that ${\overline{B}}_s[0,1]$ is $(\alpha ,\mathfrak{c})$-spaceable if and only if $\alpha <{\aleph }_0$. This completely resolves an open question raised by Liu et al. concerning the upper box dimension of graphs of continuous functions.