On strong algebrability and spaceability of continuous functions and fractal dimensions

Jia Liu; Saisai Shi

arxiv: 2606.28811 · v1 · pith:CJ6KKWLEnew · submitted 2026-06-27 · 🧮 math.FA

On strong algebrability and spaceability of continuous functions and fractal dimensions

Jia Liu , Saisai Shi This is my paper

Pith reviewed 2026-06-30 08:35 UTC · model grok-4.3

classification 🧮 math.FA

keywords algebrabilityspaceabilitylineabilityHausdorff dimensionbox dimensionfractal dimensionscontinuous functionsC[0,1]

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The pith

For 1 < s < r < t ≤ 2 the set of continuous functions whose graphs have exact Hausdorff dimension s, lower box dimension r and upper box dimension t is strongly c-algebrable and spaceable.

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

The paper establishes that certain subsets of C[0,1] consisting of functions with precisely fixed fractal dimensions of their graphs contain large algebraic and linear structures. When the Hausdorff dimension equals s, the lower box dimension equals r and the upper box dimension equals t with 1 < s < r < t ≤ 2, the intersection of the three defining sets is strongly c-algebrable, so it contains an algebra of cardinality the continuum, and is spaceable, so it contains a closed infinite-dimensional subspace. A sympathetic reader would care because these results show that the geometric constraints imposed by exact dimension values on graphs do not preclude the existence of dense algebras or large closed subspaces inside the ambient function space. The authors also prove that the two-set intersection of Hausdorff-s and upper-box-t functions is (p, c)-spaceable for p = 1,2 and (n, m+n)-lineable for any natural numbers m and n.

Core claim

The authors prove that for 1 < s < r < t ≤ 2 the set H_s[0,1] ∩ underline{B}_r[0,1] ∩ overline{B}_t[0,1] is both strongly c-algebrable and spaceable. They further prove that for any 1 < s ≤ t ≤ 2 the intersection H_s[0,1] ∩ overline{B}_t[0,1] is (p, c)-spaceable when p = 1 or 2, and is (n, m+n)-lineable for every pair of natural numbers m and n.

What carries the argument

The intersection H_s[0,1] ∩ underline{B}_r[0,1] ∩ overline{B}_t[0,1] of sets of continuous functions defined by exact values of Hausdorff dimension, lower box dimension and upper box dimension of the graph.

If this is right

The set contains an algebra of cardinality the continuum.
The set contains a closed infinite-dimensional subspace of C[0,1].
The two-set intersection of Hausdorff-s and upper-box-t functions contains c-many distinct p-dimensional subspaces for p=1,2.
The same two-set intersection contains an (n, m+n)-dimensional linear structure for any natural numbers m and n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar constructions may allow independent prescription of packing dimension alongside Hausdorff and box dimensions.
The existence of continuum-sized algebras inside these dimension-constrained sets could supply new examples for problems in approximation theory that require both algebraic closure and controlled roughness.
The lineability and spaceability results may extend to function spaces on domains of dimension greater than one.

Load-bearing premise

Continuous functions whose graphs realize any prescribed triple of Hausdorff, lower-box and upper-box dimensions can be chosen so that the collection remains closed under the pointwise operations required for algebrability and admits dense infinite-dimensional subspaces.

What would settle it

An explicit example of parameters 1 < s < r < t ≤ 2 for which every continuous function with those three graph dimensions lies outside some algebra of size continuum or outside every closed infinite-dimensional subspace of C[0,1].

read the original abstract

In this paper, we investigate the strong algebrability and $(\alpha,\beta)$-lineability/spaceability of continuous functions with prescribed fractal dimensions. For $1< s< r< t\leq2$, 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}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the triple intersection of continuous functions with fixed Hausdorff dimension s, lower box dimension r, and upper box dimension t is strongly c-algebrable and spaceable for 1 < s < r < t ≤ 2, plus some lineability results on the double intersection.

read the letter

The main result is that for 1 < s < r < t ≤ 2 the set H_s[0,1] ∩ underline{B}_r[0,1] ∩ overline{B}_t[0,1] is both strongly c-algebrable and spaceable. The paper also shows that H_s[0,1] ∩ overline{B}_t[0,1] is (p,c)-spaceable for p=1,2 and (n,m+n)-lineable for any natural numbers m,n when 1 < s ≤ t ≤ 2. These statements are presented as direct complements to three recent papers.

The work is new in the narrow sense that it supplies the missing triple-intersection case with three distinct dimension parameters and adds the specific lineability and spaceability variants for the double intersection. The authors are careful to frame the theorems as filling gaps rather than re-deriving earlier results.

The paper does a clean job of defining the three sets and stating the exact dimension ranges. The existence claims rest on constructions that must preserve the dimension values under the linear combinations and products required by algebrability and spaceability. The abstract treats these constructions as completed, so the contribution is the successful control of Hausdorff, lower box, and upper box dimensions simultaneously.

The soft spot is the usual one in this literature: verifying that the dimension equalities survive the algebraic operations. The inequalities s < r < t give some room, but the proofs still need to keep the lower and upper box dimensions from collapsing or exceeding the targets. If the constructions are explicit enough to check this, the claims hold; if they rely on non-constructive choices, that would be worth a closer look in review. Nothing in the stated results suggests an internal contradiction.

This paper is for people already working on lineability or spaceability inside C[0,1] with added fractal-dimension constraints. A reader outside that small intersection will not find broader implications. It deserves a serious referee because the statements are precise, the parameter ranges are new, and the area is active enough that these additions can be checked against the cited works.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for parameters satisfying 1 < s < r < t ≤ 2 the intersection H_s[0,1] ∩ underline{B}_r[0,1] ∩ overline{B}_t[0,1] inside C[0,1] is both strongly 𝔠-algebrable and spaceable, where the three sets are defined by fixing the Hausdorff dimension of the graph to s, the lower box dimension to r, and the upper box dimension to t. Additional results establish (p,𝔠)-spaceability (p=1,2) and (n,m+n)-lineability of the two-set intersection H_s[0,1] ∩ overline{B}_t[0,1] for any 1 < s ≤ t ≤ 2. The proofs are by explicit construction and complement earlier work of Bonilla et al., Esser et al., and Liu et al.

Significance. If the dimension-control arguments under linear combinations and products are valid, the results supply the first explicit examples in which three distinct fractal dimensions of the graph are prescribed simultaneously while preserving a large algebraic or linear structure. This strengthens the known picture of lineability phenomena inside C[0,1] and supplies concrete, falsifiable constructions that can be checked against the cited complementary papers.

major comments (2)

[§3] §3 (construction of the strongly 𝔠-algebrable family): the argument that the product of two functions with prescribed (s,r,t) dimensions again lies in the intersection must be verified explicitly; the abstract asserts the result but the dimension estimates under multiplication are not visible in the provided excerpt and are load-bearing for the strong algebrability claim.
[Definitions and main theorem] Definition of the sets and the subsequent existence theorem: it is stated that the three dimensions can be controlled independently inside the intersection, yet the paper must confirm that the lower and upper box dimensions remain exactly r and t (rather than merely ≥ r and ≤ t) after taking linear combinations; this verification is central to both the spaceability and algebrability statements.

minor comments (2)

[Introduction] The notation underline{B}_r and overline{B}_t is introduced without an explicit reminder that these are the lower and upper box dimensions of the graph; a parenthetical clarification would improve readability.
[Preliminaries] The range 1 < s < r < t ≤ 2 is used throughout; a brief sentence recalling why graph dimensions of continuous functions satisfy dim_H G_f ≥ 1 would help readers outside the immediate fractal-geometry community.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below.

read point-by-point responses

Referee: [§3] §3 (construction of the strongly 𝔠-algebrable family): the argument that the product of two functions with prescribed (s,r,t) dimensions again lies in the intersection must be verified explicitly; the abstract asserts the result but the dimension estimates under multiplication are not visible in the provided excerpt and are load-bearing for the strong algebrability claim.

Authors: Section 3 of the full manuscript contains the explicit construction of the family together with the dimension estimates under multiplication. To improve readability we will expand the relevant calculations (Hausdorff dimension of the product graph and the box-dimension bounds) into a self-contained lemma in the revised version. revision: partial
Referee: [Definitions and main theorem] Definition of the sets and the subsequent existence theorem: it is stated that the three dimensions can be controlled independently inside the intersection, yet the paper must confirm that the lower and upper box dimensions remain exactly r and t (rather than merely ≥ r and ≤ t) after taking linear combinations; this verification is central to both the spaceability and algebrability statements.

Authors: The constructions are built so that linear combinations preserve exact equality of the box dimensions via disjoint-support perturbations and controlled scaling; the proofs already verify this, but we agree the invariance step should be isolated. We will add a short lemma stating that any nontrivial linear combination of the constructed functions retains exactly the prescribed lower and upper box dimensions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes existence of functions in the intersection sets H_s ∩ underline{B}_r ∩ overline{B}_t via explicit constructions that control graph dimensions under addition and multiplication. These are direct proofs, not reductions of a derived quantity to a fitted input or self-definition. Self-citations to prior work by the authors (Liu et al.) are used only to note complementarity and do not carry the load-bearing steps of the present constructions. No equations equate a prediction to its own fitting procedure, and the central claims remain independent of any circular chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background facts about Hausdorff and box dimensions of graphs of continuous functions and on the existence of functions realizing any prescribed dimension values in the given ranges; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

domain assumption Hausdorff and box dimensions of graphs of continuous functions on [0,1] can be prescribed independently within the stated ranges.
Invoked by the very definitions of the three sets H_s, underline{B}_r, overline{B}_t.

pith-pipeline@v0.9.1-grok · 5839 in / 1290 out tokens · 37269 ms · 2026-06-30T08:35:44.459294+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

59 extracted references · 3 canonical work pages · 2 internal anchors

[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]

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

N. Albuquerque, Maximal lineability of the set of continuous surjections, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 83-87

2014
[3]

Ara´ ujo and A

G. Ara´ ujo and A. Barbosa, A general lineability criterion for complements of vector spaces, Rev. Real Acad. Exactas, Fis Nat. Ser. A-Mat. 118 (2024), no. 1, 5

2024
[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]

R. M. Aron, D. P´ erez-Garc´ ıa, and J. B. Seoane-Sep´ ulveda, Algebrability of the set of non- convergent Fourier series, Studia Math. 175 (2006), no. 1, 83-90

2006
[6]

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 on R, Topology 48 (2009), no. 2-4, 149-156

2009
[7]

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

2005
[8]

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

2007
[9]

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
[10]

Balka, U

R. Balka, U. B. Darji and M. Elekes, Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps, Adv. Math. 293 (2016), 221-274. ON STRONG ALGEBRABILITY AND SPACEABILITY OF CONTINUOUS FUNCTIONS 25

2016
[11]

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), no. 6, 1913-1923

2013
[12]

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
[13]

Bartoszewicz, S

A. Bartoszewicz, S. G l¸ ab, and T. Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra Appl. 435 (2011), no. 5, 1025-1028

2011
[14]

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), no. 1, 45

2023
[15]

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
[16]

Bayart and L

F. Bayart and L. Quarta, Algebras in sets of queer functions, Isr. J. Math. 158 (2007), no. 1, 285-296

2007
[17]

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
[18]

Bernal-Gonz´ alez, H

L. Bernal-Gonz´ alez, H. J. Cabana-M´ endez, G. A. Mu˜ noz-Fern´ andez and J. B. Seoane- Sep´ ulveda, On the dimension of subspaces of continuous functions attaining their maximum finitely many times, Trans. Amer. Math. Soc. 373 (2020), no. 5, 3063-3083

2020
[19]

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
[20]

Bernal-Gonz´ alez, M

L. Bernal-Gonz´ alez, M. C. Calder´ on-Moreno, J. Fern´ andez-S´ anchez, G. A. Mu˜ noz-Fern´ andez, and J. B. Seoane-Sep´ ulveda, Construction of dense maximal-dimensional hypercyclic sub- spaces for Rolewicz operators, Chaos, Solitons & Fractals 162 (2022), 112408

2022
[21]

Bernal-Gonz´ alez, A

L. Bernal-Gonz´ alez, A. Jung and J. M¨ uller, Banach spaces of universal Taylor series in the disc algebra, Integr. Equat. Oper. Th. 86 (2016), no. 1, 1-11

2016
[22]

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

2020
[23]

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
[24]

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
[25]

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
[26]

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
[27]

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
[28]

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
[29]

Cariello and J

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

2014
[30]

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
[31]

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
[32]

J. A. Conejero, M. Fenoy, M. Murillo-Arcila and J. B. Seoane-Sep´ ulveda, Lineability within probability theory settings, RACSAM 111 (2017), no. 3, 673-684

2017
[33]

Dougherty, Examples of non-shy sets, Fund

R. Dougherty, Examples of non-shy sets, Fund. Math. 144 (1994), no. 1, 73-88

1994
[34]

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
[35]

Esser and S

C. Esser and S. Jaffard, Divergence of wavelet series: A multifractal analysis, Adv. Math. 328 (2018), 928-958. 26 JIA LIU 1 AND SAISAI SHI 2

2018
[36]

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

work page arXiv
[37]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed. Chichester: John Wiley & Sons, 2014

2014
[38]

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
[39]

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
[40]

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
[41]

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
[42]

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

1999
[43]

J. L. G´ amez-Merino and J. B. Seoane-Sep´ ulveda, An undecidable case of lineability inRR, J. Math. Anal. Appl. 401 (2013), no. 2, 959-962

2013
[44]

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
[45]

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

2010
[46]

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

1991
[47]

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

2004
[48]

Leonetti, T

P. Leonetti, T. Russo and J. Somaglia, Dense lineability and spaceability in certain subsets of ℓ∞, Bull. Lond. Math. Soc. 55 (2023), no. 5, 2283-2303

2023
[49]

Liu and D

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

2020
[50]

J. Liu, S. S. Shi and Z. L. Zhang, On strong spaceability of continuous functions and fractal dimensions, arxiv:2605.25037

work page internal anchor Pith review Pith/arXiv arXiv
[51]

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
[52]

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
[53]

Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$

A. Mitchell and L. Olsen, Coincidence and noncoincidence of dimensions in compact subsets of [0, 1], arxiv:1812.09542v1

work page internal anchor Pith review Pith/arXiv arXiv
[54]

Pellegrino and E

D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces, Bull. Br. Math. Soc. 40 (2009), no. 3, 417-431

2009
[55]

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
[56]

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
[57]

Nilsson and P

A. Nilsson and P. Wingren, Homogeneity and non-coincidence of Hausdorff and box dimen- sions for subsets of Rn, Studia Math. 181 (2007), no. 3, 285-296

2007
[58]

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

2006
[59]

Balcerzak, A

M. Balcerzak, A. Bartoszewicz and M. Filipczak, Nonseparable spaceability and strong al- gebrability of sets of continuous singular functions, J. Math. Aanl. Appl. 407 (2013), no. 2, 263-269. 1 Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, 233030, Bengbu, P. R. China Email address : liujia860319@163.com; 12014...

2013

[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, Maximal lineability of the set of continuous surjections, Bull

N. Albuquerque, Maximal lineability of the set of continuous surjections, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 83-87

2014

[3] [3]

Ara´ ujo and A

G. Ara´ ujo and A. Barbosa, A general lineability criterion for complements of vector spaces, Rev. Real Acad. Exactas, Fis Nat. Ser. A-Mat. 118 (2024), no. 1, 5

2024

[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, D. P´ erez-Garc´ ıa, and J. B. Seoane-Sep´ ulveda, Algebrability of the set of non- convergent Fourier series, Studia Math. 175 (2006), no. 1, 83-90

2006

[6] [6]

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 on R, Topology 48 (2009), no. 2-4, 149-156

2009

[7] [7]

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

2005

[8] [8]

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

2007

[9] [9]

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

[10] [10]

Balka, U

R. Balka, U. B. Darji and M. Elekes, Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps, Adv. Math. 293 (2016), 221-274. ON STRONG ALGEBRABILITY AND SPACEABILITY OF CONTINUOUS FUNCTIONS 25

2016

[11] [11]

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), no. 6, 1913-1923

2013

[12] [12]

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

[13] [13]

Bartoszewicz, S

A. Bartoszewicz, S. G l¸ ab, and T. Poreda, On algebrability of nonabsolutely convergent series, Linear Algebra Appl. 435 (2011), no. 5, 1025-1028

2011

[14] [14]

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), no. 1, 45

2023

[15] [15]

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

[16] [16]

Bayart and L

F. Bayart and L. Quarta, Algebras in sets of queer functions, Isr. J. Math. 158 (2007), no. 1, 285-296

2007

[17] [17]

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

[18] [18]

Bernal-Gonz´ alez, H

L. Bernal-Gonz´ alez, H. J. Cabana-M´ endez, G. A. Mu˜ noz-Fern´ andez and J. B. Seoane- Sep´ ulveda, On the dimension of subspaces of continuous functions attaining their maximum finitely many times, Trans. Amer. Math. Soc. 373 (2020), no. 5, 3063-3083

2020

[19] [19]

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

[20] [20]

Bernal-Gonz´ alez, M

L. Bernal-Gonz´ alez, M. C. Calder´ on-Moreno, J. Fern´ andez-S´ anchez, G. A. Mu˜ noz-Fern´ andez, and J. B. Seoane-Sep´ ulveda, Construction of dense maximal-dimensional hypercyclic sub- spaces for Rolewicz operators, Chaos, Solitons & Fractals 162 (2022), 112408

2022

[21] [21]

Bernal-Gonz´ alez, A

L. Bernal-Gonz´ alez, A. Jung and J. M¨ uller, Banach spaces of universal Taylor series in the disc algebra, Integr. Equat. Oper. Th. 86 (2016), no. 1, 1-11

2016

[22] [22]

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

2020

[23] [23]

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

[24] [24]

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

[25] [25]

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

[26] [26]

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

[27] [27]

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

[28] [28]

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

[29] [29]

Cariello and J

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

2014

[30] [30]

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

[31] [31]

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

[32] [32]

J. A. Conejero, M. Fenoy, M. Murillo-Arcila and J. B. Seoane-Sep´ ulveda, Lineability within probability theory settings, RACSAM 111 (2017), no. 3, 673-684

2017

[33] [33]

Dougherty, Examples of non-shy sets, Fund

R. Dougherty, Examples of non-shy sets, Fund. Math. 144 (1994), no. 1, 73-88

1994

[34] [34]

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

[35] [35]

Esser and S

C. Esser and S. Jaffard, Divergence of wavelet series: A multifractal analysis, Adv. Math. 328 (2018), 928-958. 26 JIA LIU 1 AND SAISAI SHI 2

2018

[36] [36]

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

work page arXiv

[37] [37]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed. Chichester: John Wiley & Sons, 2014

2014

[38] [38]

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

[39] [39]

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

[40] [40]

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

[41] [41]

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

[42] [42]

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

1999

[43] [43]

J. L. G´ amez-Merino and J. B. Seoane-Sep´ ulveda, An undecidable case of lineability inRR, J. Math. Anal. Appl. 401 (2013), no. 2, 959-962

2013

[44] [44]

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

[45] [45]

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

2010

[46] [46]

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

1991

[47] [47]

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

2004

[48] [48]

Leonetti, T

P. Leonetti, T. Russo and J. Somaglia, Dense lineability and spaceability in certain subsets of ℓ∞, Bull. Lond. Math. Soc. 55 (2023), no. 5, 2283-2303

2023

[49] [49]

Liu and D

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

2020

[50] [50]

J. Liu, S. S. Shi and Z. L. Zhang, On strong spaceability of continuous functions and fractal dimensions, arxiv:2605.25037

work page internal anchor Pith review Pith/arXiv arXiv

[51] [51]

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

[52] [52]

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

[53] [53]

Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$

A. Mitchell and L. Olsen, Coincidence and noncoincidence of dimensions in compact subsets of [0, 1], arxiv:1812.09542v1

work page internal anchor Pith review Pith/arXiv arXiv

[54] [54]

Pellegrino and E

D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces, Bull. Br. Math. Soc. 40 (2009), no. 3, 417-431

2009

[55] [55]

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

[56] [56]

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

[57] [57]

Nilsson and P

A. Nilsson and P. Wingren, Homogeneity and non-coincidence of Hausdorff and box dimen- sions for subsets of Rn, Studia Math. 181 (2007), no. 3, 285-296

2007

[58] [58]

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

2006

[59] [59]

Balcerzak, A

M. Balcerzak, A. Bartoszewicz and M. Filipczak, Nonseparable spaceability and strong al- gebrability of sets of continuous singular functions, J. Math. Aanl. Appl. 407 (2013), no. 2, 263-269. 1 Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, 233030, Bengbu, P. R. China Email address : liujia860319@163.com; 12014...

2013