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On strong spaceability of continuous functions and fractal dimensions

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Pith's one-line read The sets of continuous functions whose graphs have fixed Hausdorff dimension s are (1, continuum)- and (2, continuum)-spaceable and (n, n+m)-lineable, while the corresponding upper box dimension sets are spaceable precisely when the space d

desk verdict They fully settle the spaceability question for upper box dimension sets and give concrete partial results for the Hausdorff case. read the letter →

arxiv 2605.25037 v2 pith:7PLDEW32 submitted 2026-05-24 math.FA

Jia Liu , Saisai Shi , Zhenliang Zhang This is my paper · ORCID
classification math.FA
keywords lineabilityspaceabilityHausdorffdimensionupperboxcontinuousfunctionsgraphfractaldimensionsC[01]
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the linear and topological size of two collections inside the space of continuous functions on [0,1]: those whose graphs have exact Hausdorff dimension s, and those whose graphs have exact upper box dimension s. It establishes that the Hausdorff-dimension collection contains closed subspaces of dimension 1 and 2 with the full space of cardinality of the continuum, plus finite-dimensional linear structures of every possible size. The upper-box-dimension collection contains such subspaces if and only if their dimension is a finite cardinal. These statements supply a partial answer for Hausdorff dimension and a complete resolution for upper box dimension to earlier questions about the algebraic structure of these dimension-constrained sets.

What carries the argument

Explicit constructions of families of functions in C[0,1] such that their linear spans and closures keep the Hausdorff or upper box dimension of every graph exactly equal to the prescribed value s.

What would settle it

An explicit nontrivial linear combination from one of the constructed families whose graph has Hausdorff dimension different from s, or a closed infinite-dimensional subspace lying inside overline B_s[0,1].

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Extended reading notes

Core claim

The authors prove that H_s[0,1] is (p, c)-spaceable for p=1,2 and also (n, n+m)-lineable for any m,n in N. Furthermore, overline B_s[0,1] is (alpha, c)-spaceable if and only if alpha < aleph_0.

Load-bearing premise

Linear combinations and closures of the constructed function families preserve the precise Hausdorff or upper box dimension value of the graph.

Editorial extensions

If this is right

  • There exist 1-dimensional and 2-dimensional closed subspaces of C[0,1] contained in H_s[0,1] union {0}.
  • For every pair of natural numbers n and m there exists an (n+m)-dimensional subspace of C[0,1] with the required lineability property inside H_s[0,1].
  • overline B_s[0,1] contains no closed infinite-dimensional subspace.
  • The spaceability threshold for the upper-box-dimension sets is exactly the first infinite cardinal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The sharp difference in allowed subspace sizes between Hausdorff and upper box dimension may reflect differing stability of these dimensions under linear operations.
  • Analogous lineability and spaceability statements could be examined for other fractal dimensions such as packing dimension.
  • The same construction technique might extend to continuous functions on domains of higher dimension.
  • If dimension preservation holds for wider classes of linear combinations, the results could constrain approximation schemes that must respect fixed graph dimension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper defines H_s[0,1] as the set of continuous functions f on [0,1] whose graph has Hausdorff dimension exactly s (s ∈ (1,2]) and ar B_s[0,1] as the analogous set for upper box dimension s. It proves that H_s[0,1] is (p, 𝔠)-spaceable for p=1,2 and (n,n+m)-lineable for all natural numbers n,m, partially answering a question of Liu et al. It further proves that ar B_s[0,1] is (α, 𝔠)-spaceable if and only if α < ℵ₀, completely resolving the corresponding open question of Liu et al.

Significance. If the dimension-preservation arguments under linear combinations and closures hold, the results supply new information on the algebraic size of level sets for graph dimensions in C[0,1]. The complete if-and-only-if characterization for the upper-box-dimension case is a substantive contribution that closes an open problem; the partial results for Hausdorff dimension add concrete positive instances of spaceability.

minor comments (2)
  1. The abstract states the main theorems but the introduction or §1 should explicitly recall the precise formulation of the open questions from Liu et al. that are being addressed, including the exact cardinal parameters involved.
  2. Notation for the graph G_f and the dimension symbols is standard, but a short preliminary subsection collecting the definitions of Hausdorff and upper box dimension (with references) would improve readability for readers outside fractal geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. We are pleased that the complete resolution of the open question for upper box dimension and the partial results for Hausdorff dimension are viewed as substantive contributions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The provided abstract and context show the paper proving new results on (α,β)-lineability/spaceability of H_s[0,1] and overline{B}_s[0,1], resolving open questions from prior work. No full derivation chain, equations, or lemmas are supplied that reduce any central claim to a self-citation, fitted input, or self-definition. The dimension-preservation assumption is stated as a hypothesis for the constructions rather than derived circularly. This is a standard case of independent mathematical work extending prior open problems, warranting score 0.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard mathematical axioms and definitions from real analysis and fractal geometry; no new entities or free parameters are introduced in the abstract.

assumptions (2)
  • standard math Hausdorff dimension and upper box dimension are well-defined for graphs of continuous functions and take values in [1,2].
    Standard properties from fractal geometry used in the definitions of the sets.
  • standard math The space C[0,1] is a Banach space over the reals.
    Background for lineability and spaceability concepts.

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Pith. "Pith review of On strong spaceability of continuous functions and fractal dimensions." pith.science (2026). https://pith.science/paper/7PLDEW32

@misc{pith2026260525037,
  author       = {Pith},
  title        = {Pith review of: On strong spaceability of continuous functions and fractal dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PLDEW32}},
  note         = {Machine review of arXiv:2605.25037}
}
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 principal 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.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On strong algebrability and spaceability of continuous functions and fractal dimensions

    math.FA 2026-06 unverdicted novelty 5.0 of 10

    Intersections of continuous functions with prescribed Hausdorff dimension s and box dimensions r,t are shown to be strongly c-algebrable and spaceable, plus related lineability results.

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Works this paper leans on

49 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aizpuru, C

    A. Aizpuru, C. P´ erez-Eslava and J. B. Seoane-Sep´ ulveda, Linear structure of sets of divergent sequences and series, Linear Algebra Appl. 418 (2006), no. 2-3, 595-598

    2006

  2. [2]

    Albuquerque, L

    N. Albuquerque, L. Bernal-Gonz´ alez, D. Pellegrino and J. B. Seoane-Sep´ ulveda, Peano curves on topological vector spaces, Linear Algebra Appl. 460 (2014), 81-96

    2014

  3. [3]

    Albuquerque, Maximal lineability of the set of continous surjections, Bull

    N. Albuquerque, Maximal lineability of the set of continous surjections, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 83-87. ON STRONG SPACEABILITY OF CONTINUOUS FUNCTIONS 29

    2014

  4. [4]

    R. M. Aron, L. Bernal-Gonz´ alez, D. M. Pellegrino, and J. B. Seoane-Sep´ ulveda, Lineability: the search for linearity in mathematics, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016

    2016

  5. [5]

    R. M. Aron , F. J. Garc´ ıa-Pacheco, D. P´ erez-Garc´ ıa and J. B. Seoane-Sep´ ulveda, On dense- lineability of sets of functions onR, Topology 48 (2009), no. 2-4, 149-156

    2009

  6. [6]

    R. M. Aron, V. I. Gurariy and J. B. Seoane-Sep´ ulveda, Lineability and spaceability of sets of functions onR, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795-803

    2005

  7. [7]

    R. M. Aron and J. B. Seoane-Sep´ ulveda, Algebrability of the set of everywhere surjective functions onC, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 1, 25-31

    2007

  8. [8]

    Balka, Dimensions of graphs of prevalent continuous maps, J

    R. Balka, Dimensions of graphs of prevalent continuous maps, J. Fractal Geom. 3 (2016), no. 4, 407-428

    2016

Show all 49 references
  1. [9]

    Balka, U

    R. Balka, U. B. Darji and M. Elekes, Haudorff and packing dimension of fibers and graphs of prevalent continuous maps, Adv. Math. 293 (2016), 221-274

    2016

  2. [10]

    C. S. B´ arroso, G. Botelho, V. V. F´ avaro and D. Pellegrino, Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces, Proc. Amer. Math. Soc. 141 (2013), 1913-1923

    2013

  3. [11]

    Bartoszewicz and S

    A. Bartoszewicz and S. G l¸ ab, Strong algebrability of sets of sequences and functions, Proc. Amer. Math. Soc. 141 (2013), no. 3, 827-835

    2013

  4. [12]

    Bartoszewicz, M

    A. Bartoszewicz, M. Filipczak and S. G l¸ ab, Algebraic structures in the set of sequences of independent random variables, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat 117 (2023), 45

    2023

  5. [13]

    Bayart, Topological and algebraic genericity of divergence and universality, Studia Math

    F. Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), no. 2, 161-181

    2005

  6. [14]

    Bayart and Y

    F. Bayart and Y. Heurteaux, On the Hausdorff dimension of graphs of prevalent continu- ous functions on compact sets, (In Further Developments in Fractals and Related Feilds, Birkh¨ auser, Boston, 2012)

    2012

  7. [15]

    Bernal-Gonz´ alez and M

    L. Bernal-Gonz´ alez and M. O. Cabrera, Lineability criteria, with applications, J. Funct. Anal. 266 (2014), no. 4, 3997-4025

    2014

  8. [16]

    Bernal-Gonz´ alez, G

    L. Bernal-Gonz´ alez, G. A. Mu˜ noz-Fern´ andez, D. L. Rodr´ ıguez-Vidanes and J. B. Seoane- Sep´ ulveda, Algebraic genericity within the class of sup-measurable functions, J. Math. Anal. Appl. 483 (2020), no. 1, 123576, 16

    2020

  9. [17]

    Bernal-Gonz´ alez, D

    L. Bernal-Gonz´ alez, D. Pellegrino and J. B. Seoane-Sep´ ulveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71-130

    2014

  10. [18]

    Bonilla, G

    A. Bonilla, G. A. Mu˜ noz-Fern´ andez, J. A. Prado-Bassas and J. B. Seoane-Sep´ ulveda, Haus- dorff and box dimensions of continuous functions and lineability, Linear & Multilinear Algebra 69 (2021), no. 4, 593-606

    2021

  11. [19]

    Botelho, D

    G. Botelho, D. Diniz, V.V. F´ avaro and D. Pellegrino, Spaceability in Banach and quasi- Banach sequence spaces, Linear Algebra Appl. 434 (2011), no. 5, 1255-1260

    2011

  12. [20]

    Botelho, D

    G. Botelho, D. Diniz and D. Pellegrino, Lineability of the set of bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 357 (2009), no. 1, 171-175

    2009

  13. [21]

    Botelho and V.V

    G. Botelho and V.V. F´ avaro, Constructing Banach spaces of vector-valued sequences with special properties, Michigan Math. J. 64 (2015), no. 3, 539-554

    2015

  14. [22]

    Cariello, V

    D. Cariello, V. V. F´ avaro and J. B. Seoane-Sep´ ulveda, Self-similar functions, fractals and algebraic genericity, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4151-4159

    2017

  15. [23]

    Cariello and J

    D. Cariello and J. B. Seoane-Sep´ ulveda, Basic sequences and spaceability inℓ p spaces, J. Funct. Anal. 266 (2014), no. 6, 3797–3814

    2014

  16. [24]

    K. C. Ciesielski and J. B. Seoane-Sep´ ulveda, A century of Sierpi´ nski-Zygmund functions, Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM 113 (2019), no. 4, 3863-3901

    2019

  17. [25]

    K. C. Ciesielski and J. B. Seoane-Sep´ ulveda, Differentiability versus continuity: restriction and extension theorems and monstrous examples, Bull. Amer. Math. Soc. (N.S.) 56 (2019), no. 2, 211-260

    2019

  18. [26]

    J. A. Conejero, M. Fenoy, M. Murillo-Arcila and J. B. Seoane-Sep´ ulveda, Lineability within probability theory settings, RACSAM 111 (2017), 673-684

    2017

  19. [27]

    P. H. Enflo, V. I. Gurariy and J. B. Seoane-Sep´ ulveda, Some results and open questions on spaceability in function spaces, Trans. Amer. Math. Soc. 366 (2014), no. 2, 611-625

    2014

  20. [28]

    Esser and S

    C. Esser and S. Jaffard, Divergence of wavelet series: A multifractal analysis, Adv. Math. 328 (2018), 928-958. 30 JIA LIU 1 SAISAI SHI2 AND ZHENLIANG ZHANG 3

    2018

  21. [29]

    Esser, S

    C. Esser, S. Maghsoudi, D. L. Rodr´ ıguez-Vidanes and J. B. Seoane-Sep´ ulveda, Algebraic structures featuring graph dimensions, H¨ older regularity, and fractional differentiability, arxiv:2602.17336

  22. [30]

    K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, second edi- tion, Chichester: John Wiley & Sons, 2003

    2003

  23. [31]

    K. J. Falconer and J. M. Fraser, The horizon problem for prevalent surfaces, Math. Proc. Cambridge Philosophical Soc. 151 (2011), no. 2, 355-372

    2011

  24. [32]

    F´ avaro, D

    V.V. F´ avaro, D. Pellegrino, A. Raposo JR. and G. Ribeiro, General Criteria for a stronger notion of lineability, Proc. Amer. Math. Soc. 152 (2024), no. 3, 941-954

    2024

  25. [33]

    V. V. F´ avaro, D. Pellegrino and D. Tom´ az, Lineability and spaceability: a new approach, Bull. Braz. Math. Soc. New Ser. 51 (2020), no. 1, 27-46

    2020

  26. [34]

    Fern´ andez-S´ anchez, J.B

    J. Fern´ andez-S´ anchez, J.B. Seoane-Sep´ ulveda and W. Trutschnig, Lineability, algebrability, and sequences of random variables, Math. Nachr. 295 (2022), no. 5, 861-875

    2022

  27. [35]

    V. P. Fonf, V. I. Gurariy and M. I. Kadets, An infinite dimensional subspace ofC[0,1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), no. 11-12, 11-16

    1999

  28. [36]

    J. L. G´ amez-Merino and J. B. Seoane-Sep´ ulveda, An undecidable case of lineability inRR, J. Math. Anal. Appl. 401 (2013), 959-962

    2013

  29. [37]

    J. L. G´ amez-Merino, G. A. Mu˜ noz-Fern´ andez, V. M. S´ anchez and J. B. Seoane-Sep´ ulveda, Sierpi´ nski-Zygmund functions and other problems on lineability, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3863-3876

    2010

  30. [38]

    J. L. G´ amez-Merino, G. A. Mu˜ noz-Fern´ andez and J. B. Seoane-Sep´ ulveda, Lineability and additivity inR R, J. Math. Anal. Appl. 369 (2010), no. 1, 265-272

    2010

  31. [39]

    V. I. Gurariy, Linear spaces composed of everywhere nondifferentiable functions (Russian), C. R. Acad. Bulgare Sci. 44 (1991), no. 5, 13-16

    1991

  32. [40]

    V. I. Gurariy and L. Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004), no. 1, 62-72

    2004

  33. [41]

    Liu and D

    J. Liu and D. Z. Liu, On the decomposition of continuous functions and dimensions, Fractals 28 (2020), no. 1, 2050007, 18

    2020

  34. [42]

    J. Liu, B. Tan and J. Wu, On the 1-prevalent continuous functions on compact sets and dimensions, Nonlinearity 36 (2023), no. 7, 3734-3750

    2023

  35. [43]

    Liu and J

    J. Liu and J. Wu, A remark on decomposition of continuous functions, J. Math. Anal. Appl. 401 (2013), no. 1, 404-406

    2013

  36. [44]

    J. Liu, Y. Zhang and S. S. Shi, On the fractal dimensions of continuous functions and algebraic genericity, J. Math. Anal. Appl. 546 (2025), no. 2, 129234

    2025

  37. [45]

    Pellegrino and E

    D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces. Bull. Braz. Math. Soc. 40 (2009), 417-431

    2009

  38. [46]

    Pellegrino and A

    D. Pellegrino and A. Raposo Jr., Pointwise lineability in sequence spaces, Indag. Math. (N.S.) 32 (2021), no. 2, 536-546

    2021

  39. [47]

    J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimen- sions, Proc. London Math. Soc. 4 (1954), no. 3, 257-302

    1954

  40. [48]

    Mu˜noz-Fern´ andez, N

    G.A. Mu˜noz-Fern´ andez, N. Palmberg, D. Puglisi and J.B. Seoane-Sep´ ulveda, Lineability in subsets of measure and function spaces, Linear Algebra Appl. 428 (2008), no. 11-12, 2805- 2812

    2008

  41. [49]

    J. B. Seoane-Sep´ ulveda, Chaos and lineability of pathological phenomena in analysis, Pro- Quest LLC, Ann Arbor, MI, Thesis (Ph.D.)–Kent State University, 2006. 1 Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, 233030, Bengbu, P. R....

    2006

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