On strong algebrability and spaceability of continuous functions and fractal dimensions

Abstract

In this paper, we investigate the strong algebrability and $(α,β)$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1< s< r< t\leq2$, we define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\},$$ $$\underline{B}_r[0,1]=\{f\in C[0,1]:\underline{\dim}_BG_f([0,1])=r\}$$ and $$\overline{B}_t[0,1]=\{f\in C[0,1]:\overline{\dim}_BG_f([0,1])=t\}.$$ We prove that $H_s[0,1]\cap\underline{B}_r[0,1]\cap\overline{B}_t[0,1]$ is both strongly $\mathfrak{c}$-algebrable and spaceable. This complements recent findings of Bonilla et al. \cite{BFBS}, Esser et al. \cite{EMVVS}, and Liu et al. \cite{LZS}. We prove that for any $1<s\leq t\leq2$, $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$. We also prove that $H_s[0,1]\cap\overline{B}_t[0,1]$ is $(n,m+n)$-lineable for any $m,n\in\mathbb{N}$, thus complementing the recent work of Liu et al. \cite{LS}.