关于连续函数的强代数性和空间性以及分形维数
On strong algebrability and spaceability of continuous functions and fractal dimensions
AI总结：
研究具有指定分形维数的连续函数的强代数性和(α,β)-线性/空间性，证明了交集集合既是强c-代数可的又是空间可的，并补充了近期结果。
摘要：
本文研究了具有指定分形维数的连续函数的强代数性和$(α,β)$-线性/空间性。对于$1< s< r< t\leq2$，我们定义$$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\},$$ $$\underline{B}_r[0,1]=\{f\in C[0,1]:\underline{\dim}_BG_f([0,1])=r\}$$ 和 $$\overline{B}_t[0,1]=\{f\in C[0,1]:\overline{\dim}_BG_f([0,1])=t\}.$$ 我们证明$H_s[0,1]\cap\underline{B}_r[0,1]\cap\overline{B}_t[0,1]$既是强$\mathfrak{c}$-代数可的又是空间可的。这补充了Bonilla等人\cite{BFBS}、Esser等人\cite{EMVVS}和Liu等人\cite{LZS}的最新发现。我们证明对于任意$1<s\leq t\leq2$，$H_s[0,1]\cap\overline{B}_t[0,1]$对于$p=1,2$是$(p,\mathfrak{c})$-空间可的。我们还证明$H_s[0,1]\cap\overline{B}_t[0,1]$对于任意$m,n\in\mathbb{N}$是$(n,m+n)$-线性可的，从而补充了Liu等人\cite{LS}的最新工作。
英文摘要：
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}.
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