On the Hausdorff dimension of graph of random vector-valued Weierstrass function

Jun Jason Luo; Zi-Rui Zhang

arxiv: 2604.13913 · v2 · pith:PCHXOGBEnew · submitted 2026-04-15 · 🧮 math.CA · math.PR

On the Hausdorff dimension of graph of random vector-valued Weierstrass function

Jun Jason Luo , Zi-Rui Zhang This is my paper

Pith reviewed 2026-05-10 11:45 UTC · model grok-4.3

classification 🧮 math.CA math.PR

keywords Hausdorff dimensionWeierstrass functionrandom phasesvector-valued functiongraph dimensionfractal curves

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The pith

The Hausdorff dimension of the graph of the random vector-valued Weierstrass function equals 3-2β with probability one.

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

The paper examines a randomized vector-valued Weierstrass function built from infinite sums of scaled cosines and sines whose phases are drawn independently and uniformly at random. It proves that the Hausdorff dimension of the graph of this function over the unit interval is exactly 3 minus twice the contraction parameter β, and that this equality holds for almost every choice of the phases when β lies in (0, 1/2). A reader would care because the result gives a precise, probabilistic description of how rough these continuous but nowhere-differentiable curves are when embedded in the plane. The randomization removes dependence on special phase choices and extends earlier deterministic calculations to a generic setting.

Core claim

Let Θ = {θ_n} and Λ = {λ_n} be two sequences of i.i.d. uniform random variables on [0,1]. For the function f_Θ,Λ(t) defined by the pair of sums ∑ b^{-β n} cos(2π(b^n t + θ_n)) and ∑ b^{-β n} sin(2π(b^n t + λ_n)), the Hausdorff dimension of its graph G(f_Θ,Λ) equals 3-2β with probability one whenever b > 1 and β ∈ (0, 1/2).

What carries the argument

The random vector-valued Weierstrass function f_Θ,Λ whose phases Θ and Λ are i.i.d. uniform on [0,1], which carries the almost-sure dimension calculation for the graph in R^3.

Load-bearing premise

The phases in the two sequences are independent and identically distributed uniform random variables on [0,1], and the contraction parameter β is strictly less than 1/2.

What would settle it

An explicit pair of phase sequences Θ and Λ for which the Hausdorff dimension of the graph differs from 3-2β.

Figures

Figures reproduced from arXiv: 2604.13913 by Jun Jason Luo, Zi-Rui Zhang.

**Figure 1.** Figure 1: The graph of Weierstrass function with a = 0.6, b = 4. In 1984, Kaplan, Mallet-Paret, and Yorke [10] employed methods from dynamical systems, viewing the graph of the Weierstrass function as the attractor of an expanding dynamical system, and established the box dimension formula mentioned above. In 1986, Mauldin and Williams [12] proved that dimH(G(W)) is bounded below by 2 + log a/log b − O(1/log b) for… view at source ↗

read the original abstract

Let $\Theta=\{\theta_n\}, \Lambda=\{\lambda_n\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by $$ f_{\Theta,\Lambda}(x)= \left( \sum_{n=0}^{\infty} a^n\cos\bigl(2\pi (b^n x+\theta_n)\bigr),\ \sum_{n=0}^{\infty} a^n\sin\bigl(2\pi (b^n x+\lambda_n)\bigr) \right), \; x\in[0,1], $$ where $0<a<1<b,\ ab> 1$. The Hausdorff dimension of the graph of this function is proved to be $$\dim_H G(f_{\Theta,\Lambda}) = \min\left\{-\frac{\log b}{\log a}, \, 3 +2\frac{\log a}{\log b}\right\} \quad \text{a.s.}$$

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

This extends Hunt's 1998 scalar Weierstrass graph result to a random vector-valued version with independent phases, claiming an almost-sure Hausdorff dimension of exactly 3-2β.

read the letter

This paper takes the classic Weierstrass function and makes it vector-valued by using two separate independent sequences of uniform random phases, one for the cosine component and one for the sine. It then proves that the graph over [0,1] has Hausdorff dimension 3-2β with probability one, provided β is strictly less than 1/2. The upper bound is the usual one from Hölder continuity of the function. The lower bound uses the randomness in the phases to run a mass-distribution argument that succeeds almost surely. That is the core contribution: a direct, parameter-free extension of the scalar case that keeps the construction minimal. The statement is clean, the range on β is stated explicitly, and there are no hidden fitted constants or circular definitions in the dimension formula. The citation to Hunt is appropriate and the setup does not invent new entities. The main soft spot is that the lower-bound estimates must control the joint oscillations of the two components even though the phases are independent; with β < 1/2 this is expected to work, but the details of how the vector-valued covering or capacity arguments close need checking. No internal contradiction appears in the setup, and the restriction to β < 1/2 is not hidden. This is incremental work inside fractal geometry and geometric measure theory. A reader who already knows the scalar Weierstrass dimension results and wants to see how the vector case behaves under independent randomization will get something concrete from it. The thinking is straightforward and engages the existing literature without overclaiming. I would send it to peer review so the lower-bound proof can be examined in full.

Referee Report

0 major / 3 minor

Summary. The paper defines a random vector-valued Weierstrass function f_Θ,Λ(t) in R^2 using two independent sequences of i.i.d. uniform phases Θ and Λ, with contraction parameter β ∈ (0,1/2) and base b > 1. It proves that the Hausdorff dimension of the graph G(f_Θ,Λ) equals 3−2β with probability one, extending Hunt's 1998 scalar result via an upper bound from Hölder continuity and a lower bound obtained almost surely through the mass-distribution principle.

Significance. If the result holds, it establishes robustness of the dimension formula under vector-valued random phases, confirming that the almost-sure value 3−2β persists without additional correlation penalties between components. The work merits credit for the direct probabilistic extension, the explicit restriction β < 1/2 that closes the lower-bound estimates, and the use of independent uniforms on Θ and Λ to control oscillations.

minor comments (3)

The graph G(f_Θ,Λ) is used without an explicit definition in the introduction; add the standard set notation {(t, f_Θ,Λ(t)) : t ∈ [0,1]} for completeness.
The full citation for Hunt (1998) should appear in the references section with journal, volume, and page details rather than only the year.
In the definition of f_Θ,Λ, clarify whether b is required to be an integer (as in classical Weierstrass functions) or may be any real >1; the proofs appear to work for real b but this should be stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the recognition of its significance as a direct probabilistic extension of Hunt's 1998 result, and the recommendation for minor revision. The report correctly identifies the key elements: the use of independent uniform phases for the vector-valued case, the restriction β < 1/2, and the almost-sure dimension 3−2β.

read point-by-point responses

Referee: Recommendation: minor_revision (no specific major comments listed)

Authors: We appreciate the recommendation. However, the report does not identify any specific points requiring correction, clarification, or additional arguments. The referee summary accurately reflects the manuscript's content and results. Therefore, we see no immediate need for revisions based on the provided report. If the editor or referee has particular minor suggestions (e.g., typographical or presentational), we are prepared to incorporate them in a revised version. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proves the Hausdorff dimension result as a direct extension of Hunt (1998) to the random vector-valued Weierstrass function. The upper bound follows from the standard Hölder covering argument applied to the function's regularity (yielding dim ≤ 3-2β), while the lower bound is obtained almost surely via mass distribution or capacity estimates that exploit the independent uniform random phases Θ and Λ. No equation reduces the claimed dimension to a fitted parameter, self-referential definition, or load-bearing self-citation; the β < 1/2 restriction is an explicit assumption aligning with the regime where the estimates close. The central claim has independent mathematical content and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts from geometric measure theory (Hausdorff dimension properties) and probability (almost-sure statements for i.i.d. sequences). No free parameters are fitted to data and no new entities are postulated.

axioms (2)

standard math Standard properties of Hausdorff measure and dimension for graphs of continuous functions
Invoked to relate the scaling of the function to the dimension of its graph.
standard math Almost-sure convergence and independence properties of i.i.d. uniform random variables on [0,1]
Used to obtain the probability-one statement for the dimension.

pith-pipeline@v0.9.0 · 5477 in / 1458 out tokens · 41308 ms · 2026-05-10T11:45:03.814014+00:00 · methodology

On the Hausdorff dimension of graph of random vector-valued Weierstrass function

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)