On strong algebrability and spaceability of continuous functions and fractal dimensions

Abstract

In this paper, we investigate the strong algebrability and $(\alpha ,\beta )$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1<s<r<t\le 2$, we define $$H_s[0,1]=\{f\in C[0,1]:{\dim }_H{G}_f([0,1])=s\},$$ $${\underline{B}}_r[0,1]=\{f\in C[0,1]:{\underline{\dim }}_B{G}_f([0,1])=r\}$$ and $${\overline{B}}_t[0,1]=\{f\in C[0,1]:{\overline{\dim }}_B{G}_f([0,1])=t\}.$$ 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 $1<s\le t\le 2$, $H_s[0,1]\cap {\overline{B}}_t[0,1]$ 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}.

27 pages