Abstract:Given $s\in(1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\}$$ and $$\overline{B}_s[0,1]=\{f\in C[0,1]:\overline{\dim}_BG_f([0,1])=s\}.$$ 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 principle 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.
| Subjects: | Functional Analysis (math.FA) |
| MSC classes: | 26A16, 28A78, 15A03, 46B87 |
| Cite as: | arXiv:2605.25037 [math.FA] |
| (or arXiv:2605.25037v1 [math.FA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25037 arXiv-issued DOI via DataCite (pending registration) |
From: Zhenliang Zhang [view email]
[v1]
Sun, 24 May 2026 12:25:01 UTC (23 KB)
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