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PreprintCC BY 4.0arXiv

(α,c)-spaceability of continuous functions with prescribed Hausdorff graph dimension

  • Anselmo Raposo Junior
  • Daniel Pellegrino
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Abstract

Let 1<s≤2 and let Hs[0,1] consist of the real-valued continuous functions on [0,1] whose graphs have Hausdorff dimension s. We prove that every finite-dimensional subspace E⊂ C[0,1] with E∖{0}⊂ Hs[0,1] is contained in a closed complemented subspace isomorphic to c0, all of whose nonzero elements still belong to Hs[0,1]. This settles a question of Liu, Shi and Zhang. We also obtain the exact criterion: Hs[0,1] is (α,c)-spaceable precisely when α is finite. We also show that every admissible finite-dimensional subspace is contained in a dense subspace of Hamel dimension c whose nonzero elements have the same prescribed graph dimension. This also recovers maximal pointwise dense-lineability of Hs[0,1].

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Cite

BibTeXRIS
BibTeX
@misc{junior2026αcspaceability,
  title = {{$(α,\mathfrakc)$-spaceability of continuous functions with prescribed Hausdorff graph dimension}},
  author = {Anselmo Raposo Junior and Daniel Pellegrino},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15992},
  eprint = {2610.15992},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15992},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - (α,c)-spaceability of continuous functions with prescribed Hausdorff graph dimension
AU  - Anselmo Raposo Junior
AU  - Daniel Pellegrino
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15992
UR  - https://arxiv.org/abs/2610.15992
AB  - Let 1<s≤2 and let H_s[0,1] consist of the real-valued continuous functions on [0,1] whose graphs have Hausdorff dimension s. We prove that every finite-dimensional subspace E⊂ C[0,1] with E∖{0}⊂ H_s[0,1] is contained in a closed complemented subspace isomorphic to c_0, all of whose nonzero elements still belong to H_s[0,1]. This settles a question of Liu, Shi and Zhang. We also obtain the exact criterion: H_s[0,1] is (α,c)-spaceable precisely when α is finite. We also show that every admissible finite-dimensional subspace is contained in a dense subspace of Hamel dimension c whose nonzero elements have the same prescribed graph dimension. This also recovers maximal pointwise dense-lineability of H_s[0,1].
ER  -