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On strong algebrability and spaceability of continuous functions and fractal dimensions

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read 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.

desk verdict 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 →

arxiv 2606.28811 v1 pith:CJ6KKWLE submitted 2026-06-27 math.FA

Jia Liu , Saisai Shi This is my paper · ORCID
classification math.FA
keywords algebrabilityspaceabilitylineabilityHausdorffdimensionboxfractaldimensionscontinuousfunctionsC[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

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.

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.

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].

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Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
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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

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)
  1. [§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.
  2. [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)
  1. [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.
  2. [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

  1. 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

  2. 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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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.

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Pith. "Pith review of On strong algebrability and spaceability of continuous functions and fractal dimensions." pith.science (2026). https://pith.science/paper/CJ6KKWLE

@misc{pith2026260628811,
  author       = {Pith},
  title        = {Pith review of: On strong algebrability and spaceability of continuous functions and fractal dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CJ6KKWLE}},
  note         = {Machine review of arXiv:2606.28811}
}
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}.

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