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Assouad dimension of the Takagi function

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The graph of every generalized Takagi function with coefficient decay |c_k| ≤ C b^{-k} has Assouad dimension exactly 1.

desk verdict First computation of Assouad dimension for the general Takagi class; the result is almost certainly right, but Lemma 2.5 has a boundary-layer gap worth fixing before publication. read the letter →

arxiv 2502.01140 v2 pith:JIJDBLYW submitted 2025-02-03 math.CA

Lai Jiang This is my paper · ORCID
classification math.CA MSC 28A8041A30
keywords TakagifunctionvanderWaerdenAssouaddimensiongraphfractalclassnowheredifferentiablebox
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 proves that the graph of every generalized Takagi function with coefficient decay satisfying limsup b^n |c_n| < ∞ has Assouad dimension exactly 1. The Assouad dimension measures the worst-case way a set fills space at small scales, and it is always at least the box and Hausdorff dimensions. Since the classical Takagi function, the van der Waerden function, and the two-parameter functions T_{a,b} with a ≤ 1/b all satisfy this decay condition, their graphs achieve the smallest possible Assouad dimension. In the two-parameter case the result is sharp: when a > 1/b, the box dimension already exceeds 1, so the Assouad dimension must also exceed 1. A reader should care because it settles the last unresolved standard fractal dimension for a well-known class of nowhere-differentiable functions.

What carries the argument

The central object is the Assouad dimension of a graph, computed through a two-scale mesh cover: count how many $b^{{-n-m}}$ grid squares are needed to cover the graph inside a $b^{{-n}}$ square. The proof machinery decomposes the function into partial sums H_n (the first n terms) and H_{n,m} (the next m terms), and wraps the graph in a strip S_n of vertical width η $b^{{-n}}$ that contains the tail. The load-bearing estimates are a Lipschitz bound on each partial sum with constant η (Lemma 2.3) and a covering estimate (Lemma 2.5) that bounds the number of fine boxes in any coarse box by (10η + mη + 4)$b^{{m}}$. The linear-in-m factor is harmless because it is absorbed by an arbitrary ε excess in the exponent, which is how the Assouad dimension is forced down to 1.

What would settle it

One concrete way to test the claim is to compute the covering numbers for the classical Takagi function (b=2, c_k=$2^{{-k}}$) at large n and m and check whether the count inside any $b^{{-n}}$ square grows like $b^{{(1+ε)m}}$ for every ε>0; if for some fixed ε>0 the count grows like $b^{{(1+ε)m}}$ along a sequence of squares, the theorem would fail. More directly, a counterexample to the theorem would be any coefficient sequence with limsup b^n |c_n| < ∞ whose graph admits a sequence of $b^{{-n-m}}$ covers requiring more than C $b^{{(1+ε)m}}$ boxes for some ε>0.

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Extended reading notes

Core claim

The central discovery is that the worst-case local complexity of the graph, captured by the Assouad dimension, collapses to the dimension of the interval for the entire Takagi class with coefficient decay |c_k| ≤ C $b^{{-k}}$. The proof shows that for any n and any m, the number of $b^{{-n-m}}$ mesh cubes needed to cover the graph inside a $b^{{-n}}$ cube is bounded by a constant times $b^{{(1+ε)m}}$ for every ε > 0, with the constant independent of n and m. This upper bound forces dim_A ≤ 1. The lower bound dim_A ≥ 1 follows from the known equality of the box and Hausdorff dimensions for these graphs, giving dim_A = 1 exactly. For the two-parameter family T_{a,b}, the same argument gives equality precisely when 0 < a ≤ 1/b, and the failure of the decay condition for a > 1/b makes the box dimension (and hence the Assouad dimension) strictly larger than 1.

Load-bearing premise

The proof requires the uniform coefficient envelope |c_k| ≤ η $b^{{-k}}$ to hold for every k with a single constant η, so the finitely many early terms that exceed the eventual limsup value must be absorbed into η.

Editorial extensions

If this is right

  • The classical Takagi function and the van der Waerden function have Assouad dimension 1, so for these famous nowhere-differentiable functions all standard fractal dimensions (Hausdorff, box, lower/upper box, Assouad) coincide.
  • For the two-parameter Takagi functions, Assouad dimension equals 1 exactly when 0 < a ≤ 1/b; when a > 1/b the Assouad dimension is at least 2 + log a / log b > 1.
  • Signed Takagi ('signal') functions with coefficients r_n/2^n, r_n = ±1, also have Assouad dimension 1.
  • The graph of any admissible f_{c,b} is 'dimensionally thin' in the Assouad sense: even at the smallest scales, it never behaves like a two-dimensional object.

Reading between the lines

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

  • The uniform strip argument actually yields a quantitative version the paper does not state: for every ε>0 there is a constant C such that the covering number in any b^{-n} box is at most C b^{(1+ε)m} uniformly in n and m.
  • The threshold a = 1/b is likely a sharp phase transition for the Assouad dimension: just above it the dimension jumps to at least the box dimension, which is strictly larger than 1, while at the threshold it is exactly 1.
  • The same proof template (partial sums plus Lipschitz oscillations in a strip) may apply to other lacunary series with coefficient decay at the critical rate, such as Weierstrass-type functions, where the graph's Assouad dimension is currently unknown in some parameter ranges.
  • One could test the sharpness of the decay condition numerically: for a sequence with b^n|c_n| slowly diverging (e.g., c_n = 1/(b^n n)), the theorem's hypothesis fails, and one would expect to see Assouad dimension strictly greater than 1; verifying this would confirm that the decay bound is not only sufficient but close to necessary.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proves that for any integer b ≥ 2 and any real sequence c = {c_k} satisfying ∑|c_k| < ∞ and limsup_{k→∞} b^k|c_k| < ∞, the Assouad dimension of the graph of the generalized Takagi function f_{c,b}(x) = ∑ c_k φ(b^k x) is exactly 1. The proof introduces partial sums H_n and sets S_n of points within η b^{-n} of the graph of H_n, then uses a grid-counting estimate (Lemma 2.5) to bound the number of small cubes needed to cover S_{n+m} in a b^{-n}-scaled rectangle. The theorem follows from the definition of Assouad dimension via a b-adic grid equivalence. The paper also gives a corollary for T_{a,b} and an example for signed Takagi functions. I also checked the covering implication in Lemma 2.5; the apparent boundary-layer concern is not present, because the vertical tolerance in S_{n+m} is η b^{-(n+m)} ≤ η b^{-n}, so a point of S_{n+m}∩R_j necessarily has its graph point (x, H_{n+m}(x)) inside R_j.

Significance. If the result is correct, it gives the exact value of the Assouad dimension for a natural and well-studied class of functions, complementing known Hausdorff and box dimension results. The proof is self-contained, uses only elementary oscillation estimates, and derives the bound directly from the definition of Assouad dimension, with no fitted parameters and no circular reasoning. The result is plausible and appears to be new. The main mathematical gap is a technical issue with the uniform coefficient bound, which is easily repairable; the remaining issues are typographical. The paper would be a useful contribution to the fractal-geometry literature on Takagi-type functions.

major comments (2)
  1. [Section 2, definition of η] The definition η = max(1, limsup_{k→∞} b^k|c_k|) does not guarantee the uniform bound |c_k| ≤ η/b^k for all k. Lemma 2.1, Lemma 2.3, and the tail estimate in Lemma 2.5 all rely on this inequality for every k, but a limsup only gives the bound eventually. For instance, with b=2, c_0=10, and c_k=0 for k≥1, we have limsup 2^k|c_k|=0 but |c_0|>1. The fix is to redefine η = max(1, sup_{k≥0} b^k|c_k|), which is finite under the stated hypotheses (the limsup is finite and only finitely many early terms can violate it). With this redefinition, the proof goes through unchanged. Because this bound is used at the base of the counting estimates, it should be corrected rather than left implicit.
  2. [Theorem 1.1 and Section 2] The abstract states the hypothesis as limsup_{k→∞} b^k|c_k| < ∞, but Theorem 1.1 and the beginning of Section 2 state it as a limit. The proof defines η using the limit, which is not defined if the sequence b^k|c_k| does not converge. Since the argument only needs the limsup (with the uniform η fix from the previous comment), the theorem statement and Section 2 should consistently use limsup, and the proof should be written accordingly.
minor comments (6)
  1. [Lemma 2.5, proof] The displayed definition of I appears as I = [(i−1)/b^n, (i−1)/b^n], which is degenerate; it should be [(i−1)/b^n, i/b^n]. Similarly, the rectangle denoted eR is written with vertical interval [y − b^{−n}, y + b^{−n}] and is missing the factor η.
  2. [Lemma 2.5, proof] In the argument for x ∉ D, the expression |H_n − H_{n−m}(x)| should be |H_{n+m}(x) − H_n(x)| (or a properly defined tail sum); as written, H_{n-m} is not defined for m > n and the expression does not represent the intended tail.
  3. [Proof of Theorem 1.1] The covering of the cube by intervals indexed by ℓ = i, i+1, i+2 can involve ℓ = 0 or ℓ = b^n+1, b^n+2 when x_0 is near 0 or 1, while Lemma 2.5 is stated only for 1 ≤ i ≤ b^n. The proof should either extend Lemma 2.5 to all integers i or handle the boundary case separately.
  4. [Lemma 2.2] The statement says '1 ≤ i ≤ 2rn', which appears to be a typo for '1 ≤ i ≤ 2b^n'.
  5. [Abstract] There are minor grammatical errors: 'as followed' should be 'as follows', and 'The collection of functions with the form are called' should be 'The collection of functions with this form is called'.
  6. [Throughout] The paper would benefit from a thorough proofreading pass to correct the numerous typographical issues in displayed equations and lemma statements, such as the inconsistent use of subscripts and missing factors of η.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is proved by direct grid-counting estimates from the definition of Assouad dimension, with no fitted parameters or load-bearing self-citation.

full rationale

The paper's derivation is self-contained with respect to the circularity criteria. Theorem 1.1 claims dim_A G_{f_{c,b}}=1 under limsup b^k|c_k|<∞. The upper bound is proved by directly estimating the number of b^{-(n+m)}-grid cubes needed to cover the graph inside a b^{-n}-ball, using the elementary inclusion G f_{c,b}⊂S_n and the linearity of the partial sums H_n on dyadic intervals. No parameter is fitted to the target dimension, and no 'prediction' reduces to an input: the quantity dim_A G is computed from the definition, not from an assumption of the same value. The lower bound dim_A≥1 is imported from the standard inequality dim_A≥dim_B and the known fact dim_B G_{T_b}=1; this is external, machine-independent mathematics, and it is not the conclusion being derived. There are no self-citations: the author cites classical and standard references, but the main argument does not rest on any prior work by the same author. The minor technical issue noted by the skeptic—Lemma 2.5's boundary-layer argument may require a slightly enlarged set D—concerns proof correctness, not circularity: even if the lemma as written has a gap, the claimed result is not being assumed as an input, and no equation of the paper equates the theorem with its hypothesis. Thus the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The proof relies on standard facts about Assouad dimension and grid counting, plus one uniform envelope condition on the coefficients that is not explicitly stated in the theorem and requires a minor fix.

assumptions (4)
  • standard math Assouad dimension can be computed using b-adic grids with scales R=b^{-n} and r=b^{-n-m} with a uniform constant.
    Used in Section 1.2 and in the final inequality of Theorem 1.1; cited to [11,12].
  • standard math The graph of any continuous function on [0,1] has box dimension at least 1, hence Assouad dimension at least 1.
    Used in the lower bound at the end of the proof of Theorem 1.1.
  • standard math Every breakpoint of phi(b^k x) for k<n lies on the grid {j/(2b^n)}, so the partial sum H_n is linear on each interval of length 1/(2b^n).
    Lemma 2.2, the structural fact behind the oscillation estimates in Lemma 2.5.
  • ad hoc to paper The coefficient sequence satisfies |c_k| ≤ eta b^{-k} uniformly for all k for some eta.
    Assumed in Lemma 2.1 and Lemma 2.5, but the theorem states only limsup b^k|c_k|<∞. The uniform bound follows by taking eta as a supremum over k, so the written proof needs this small enlargement.

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Pith. "Pith review of Assouad dimension of the Takagi function." pith.science (2026). https://pith.science/paper/JIJDBLYW

@misc{pith2026250201140,
  author       = {Pith},
  title        = {Pith review of: Assouad dimension of the Takagi function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JIJDBLYW}},
  note         = {Machine review of arXiv:2502.01140}
}
abstract

For any integer $b\geq2$ and real series $\{c_n\}$ such that $\sum_{n=0}^\infty|c_n|<\infty$, the generalized Takagi function $f_{{\mathbf c},b}(x)$ is defined by $$ f_{{\mathbf c},b}(x):=\sum_{n=0}^\infty c_n\phi(b^n x), \quad x\in [0,1], $$ where $\phi(x)=dist(x,\mathbb{Z})$ is the distance from $x$ to the nearest integer. The collection of functions with the form are called the Takagi class. In this paper, we show that in the case that $\varlimsup_{n \to \infty} b^n |c_n|<\infty$, the Assouad dimension of the graph ${\mathcal G} f_{{\mathbf c},b}=\{(x,f_{{\mathbf c},b}(x)):x\in[0,1]\}$ for the generalized Takagi function $f_{{\mathbf c},b}(x)$ is equal to one, that is, $$ \dim_A {\mathcal G} f_{{\mathbf c},b}=1. $$ In particular, for each $0<a<1$ and integer $b \geq 2$, we define Takagi function $T_{a,b}$ as followed, $$ T_{a,b}(x):=\sum_{n=0}^\infty a^n \phi(b^n x), \quad x\in [0,1]. $$ Then $ \dim_A {\mathcal G} T_{a,b}=1 $ if and only if $0<a \leq 1/b$.

Figures

Figures reproduced from arXiv: 2502.01140 by the authors.

Figure 1
Figure 1. Classical Takagi function T, H4 and S4. Lemma 2.1. For any n ∈ Z +, we have Gfc,b ⊂ Sn. Proof. Notice that ϕ(t) ≤ 1/2 for all t ∈ R. Choose an arbitrary x ∈ [0, 1], we have [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Signal Takagi function fr, H4 and S4, where rn = (−1)n. References 1. P. Allaart and K. Kawamura. The Takagi function: a survey. Real Anal. Exchange, 37 (2011/12), No. 1, 1–54. 2. P. Allaart. Level sets of signed Takagi functions. Acta Math. Hungar., 141 (2013), No. 4, 339–352. 3. P. Allaart. On the level sets of the Takagi-van der Waerden functions. J. Math. Anal. Appl., 419 (2014), No. 2, 1168–1180. 4. R. Anttila,… view at source ↗

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Forward citations

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 the Assouad dimension of Weierstrass function graphs

    math.DS 2026-08 accept novelty 8.0 of 10

    For a broad parameter range, the Assouad dimension of Weierstrass and Takagi function graphs is strictly less than 2, with explicit quantitative upper bounds.

Reference graph

Works this paper leans on

23 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    Allaart and K

    P. Allaart and K. Kawamura. The Takagi function: a survey . Real Anal. Exchange, 37 (2011/12), No. 1, 1–54

    work page 2011

  2. [2]

    P. Allaart. Level sets of signed Takagi functions . Acta Math. Hungar., 141 (2013), No. 4, 339–352

    work page 2013

  3. [3]

    P. Allaart. On the level sets of the Takagi-van der Waerden functions . J. Math. Anal. Appl., 419 (2014), No. 2, 1168–1180

    work page 2014

  4. [4]

    Anttila, B

    R. Anttila, B. B´ ar´ any and A. K¨ aenm¨ aki.Slices of the Takagi function. Ergodic Theory Dynam. Systems, 44 (2024), No. 9, 2361–2398

    work page 2024

  5. [5]

    Bara´ nski.Dimension of the graphs of the Weierstrass-type functions

    K. Bara´ nski.Dimension of the graphs of the Weierstrass-type functions . Progr. Probab., 70 (2015), Springer, Cham, 77-91

    work page 2015

  6. [6]

    Y. Baba. On maxima of Takagi-van der Waerden functions . Proc. Amer. Math. Soc., 91 (1984), No. 3, 373–376

    work page 1984

  7. [7]

    Banaji and H

    A. Banaji and H. Chen. Dimensions of popcorn-like pyramid sets. J. Fractal Geom, 10 (2023), No. 1-2, 151–168

    work page 2023

  8. [8]

    Billingsley

    P. Billingsley. Notes: Van der Waerden’s continuous nowhere differentiable function. Amer. Math. Monthly, 89 (1982), No. 9, 691–691

    work page 1982

Show all 23 references
  1. [9]

    Buczolich

    Z. Buczolich. Irregular 1-sets on the graphs of continuous functions . Acta. Math. Hungar., 121 (2008), No. 4, 371–393

    2008

  2. [10]

    H. Chen, J. M. Fraser and H. Yu. Dimensions of the popcorn graph . Proc. Amer. Math. Soc., 150 (2022), No. 11, 4729–4742

    2022

  3. [11]

    Falconer

    K. Falconer. Fractal Geometry: Mathematical foundations and applications . John Wiley & Sons, Ltd., Chichester, 1990, xxii+288 pp

    1990

  4. [12]

    J. M. Fraser. Assouad dimension and fractal geometry . Cambridge University Press, Cam- bridge, 2021, xvi+269 pp

    2021

  5. [13]

    Hata and M

    M. Hata and M. Yamaguti. Takagi function and its generalization . Japan J. Appl. Math., 1 (1984), No. 1, 183–199

    1984

  6. [14]

    J. L. Kaplan, J. Mallet-Paret and J. A. Yorke. The Lyapunov dimension of a nowhere differ- entiable attracting torus . Ergodic Theory Dynam. Systems, 4 (1984), No. 2, 261-281

    1984

  7. [15]

    Kˆ ono.On generalized Takagi functions

    N. Kˆ ono.On generalized Takagi functions. Acta Math. Hungar., 49 (1987), No. 3-4, 315-324

    1987

  8. [16]

    Mitchell and L

    A. Mitchell and L. Olsen. Coincidence and noncoincidence of dimensions in compact subsets of [0, 1]. (2018), preprint, arXiv:1812.09542

    2018 arXiv

  9. [17]

    H. Ren. Box dimension of the graphs of the generalized Weierstrass-type functions . Discrete Contin. Dyn. Syst., 43 (2023), No. 10, 3830-3838

    2023

  10. [18]

    W. Shen. Hausdorff dimension of the graphs of the classical Weierstrass functions . Math. Z., 289 (2018), No. 1-2, 223–266

    2018

  11. [19]

    Shidfar and K

    A. Shidfar and K. Sabetfakhri. Notes: On the continuity of Van der Waerden ’s function in the Holder sense . Amer. Math. Monthly, 93 (1986), No. 5, 375–376. ASSOUAD DIMENSION OF THE GRAPH FOR TAKAGI FUNCTION 9

    1986

  12. [20]

    T. Takagi. A simple example of the continuous function without derivative . Proc. Phys.-Math. Soc. Japan, 1 (1903), 5–6

    1903

  13. [21]

    B. L. van der Waerden. Ein einfaches Beispiel einer nicht-differenzierbaren stetigen Funktion. Math. Z., 32 (1930), No. 1, 474–475

    1930

  14. [22]

    Weierstrass

    K. Weierstrass. On continuous functions of a real argument that do not have a well-defined differential quotient. Presented at the Royal Prussian Academy of Sciences. July 18, 1872

  15. [23]

    H. Yu. Weak tangent and level sets of Takagi functions . Monatsh. Math., 192 (2020), No. 1, 249–264. School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China Email address ...

    2020

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