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On the Assouad dimension of Weierstrass function graphs

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Pith's one-line read The paper proves that the Assouad dimension of classical Weierstrass function graphs is strictly less than 2 for all a ∈ (0,1) and integer b with ab ≥ π+1, disproving the prevailing expectation that it should equal 2, and gives…

desk verdict First strict upper bound for the Assouad dimension of Weierstrass graphs, dim_A < 2 for ab >= pi+1, with a largely sound proof and only minor gaps in the explicit Takagi bound. read the letter →

arxiv 2608.11145 v1 pith:54KKBFAY submitted 2026-08-11 math.DS math.CAmath.MG

Efstathios Konstantinos Chrontsios Garitsis This is my paper · ORCID
classification math.DSmath.CAmath.MG MSC 28A8031E05
keywords AssouaddimensionWeierstrassfunctionsTakagiAikawaporosityfractalgraphspositivedifferenceconditionhorizontalslices
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 aims to settle the Assouad dimension of Weierstrass function graphs, which had been an open problem with many signs pointing toward the value 2. Under the condition ab ≥ π+1 it proves dim_A G(W_{a,b}) < 2 for the classical Weierstrass function, and more generally proves dim_A G(W^φ_{a,b}) ≤ 2−θ_φ < 2 for any Lipschitz 1-periodic generator φ satisfying a b-adic positive difference condition, with θ_φ explicit. The same result gives dim_A < 2 for Takagi function graphs when ab ≥ 3, contradicting a conjecture stated in [42]. The proof runs through the equivalence between the Aikawa and Assouad dimensions, counting the b-adic intervals at each scale that lie close to the graph and converting the count into an integral bound. If correct, the paper delivers the first non-trivial upper bound for these graphs, along with porosity and analytic corollaries such as Muckenhoupt and Hardy-type inequalities.

What carries the argument

The load-bearing mechanism is the b-adic positive difference (PD) condition: some truncation P_{n_φ} has a b-adic secant satisfying |P_{n_φ}(t_0+k_φ $b^{{−n_φ}}$) − P_{n_φ}(t_0)| > κ_φ k_φ $b^{{−n_φ}}$, meaning the secant is steeper than the maximal slope κ_φ that any combination of lower-frequency terms can contribute. Because φ is 1-periodic, the same strict separation persists in every longer truncation on an interval I_φ. The proof then counts the b-adic intervals at scale $b^{{−n}}$ lying close to the graph (the 'Aikawa intervals'), showing at most C $b^{{(1−θ_φ)(N−p)}}$ of them occur inside any level-p interval, and converts this count, through the known equivalence between Aikawa and Assouad dimensions, into an integral estimate on dist(z,G)^{−ϵ} that forces dim_A ≤ 2−θ_φ. Separately, a simple sufficient criterion, Lip(φ)/(ab−1) < 2 osc φ/(1+a), verifies the (PD) condition for the cosine and sawtooth generators.

What would settle it

Compute the local covering exponent of G(W_{a,b}) for a specific parameter pair in the range, say a=0.95 and b=5: if as r→0 the number of r-balls needed to cover a ball of radius R on the graph grows like (R/r)^2, then dim_A = 2 and Theorem 1.1 would be false. As a preliminary check, the one-line (PD) inequality for this pair, 2π/(ab−1) < 4/(1+a), can be evaluated directly to confirm the proof's premise holds.

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Extended reading notes

Core claim

On the paper's own terms: for a ∈ (0,1), b ∈ N with ab > 1 and a Lipschitz 1-periodic φ, define W^φ_{a,b} = Σ_{j=0}^∞ a^j φ(b^j x). If some finite truncation P_{n_φ} has a b-adic secant slope strictly larger than κ_φ = Lip(φ)/(ab−1), persisting across an interval I_φ, then dim_A G(W^φ_{a,b}) ≤ 2−θ_φ < 2, where θ_φ = −log(1−|I_φ|/2)/(L log b) for a suitable level L. For φ(t) = cos(2πt) the condition holds when ab ≥ π+1, giving dim_A G(W_{a,b}) < 2; for the sawtooth φ(t) = dist(t,Z) it holds when ab ≥ 3, giving dim_A G(T_{a,b}) < 2. The proof also yields dim_A W_y ≤ 1−θ_φ for every horizontal slice and porosity of the Weierstrass graph, with consequences for Muckenhoupt weights and Hardy inequalities.

Load-bearing premise

Everything rests on the existence of a b-adic interval where a finite truncation has a genuinely steeper slope than the maximum slope all lower-frequency terms can contribute; if no such interval exists for a given generating function, the counting estimates and the dimension bound do not apply.

Editorial extensions

If this is right

  • For every a ∈ (0,1) and integer b with ab ≥ π+1, the classical Weierstrass graph has Assouad dimension strictly below 2, answering the open question posed in [21] for this parameter range.
  • Takagi function graphs have Assouad dimension strictly below 2 whenever ab ≥ 3, refuting the conjecture in [42] that the dimension equals 2 in that range.
  • Every horizontal slice W_y = {x ∈ [0,1] : W^φ_{a,b}(x) = y} satisfies dim_A W_y ≤ 1−θ_φ < 1, so all slices are porous subsets of R.
  • The Weierstrass graph is porous (Corollary 1.5), and as a result the distance weight dist(z,G)^{−α} belongs to the Muckenhoupt class A_p exactly when (1−p)(2−dim_A G) < α < 2−dim_A G; related Triebel–Lizorkin Hardy inequalities hold for sp < θ_φ.
  • The bounds are quantitative: for the cosine generator, dim_A G(W_{a,b}) ≤ 2 − log(13/12)/(L log b) with L given explicitly, and for Takagi functions a similar bound uses log(25/23).

Reading between the lines

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

  • The (PD) condition is a steep-secant hypothesis; a natural guess is that any generating function whose scaled increments are genuinely one-sided at b-adic scales should satisfy it, which would extend the theorem beyond cosine and sawtooth to, for example, random or generic Lipschitz phases.
  • Because θ_φ is built from the length of the interval I_φ, optimizing the choices of n_φ and k_φ could yield dimension bounds closer to the true value; the paper does not claim sharpness, so the explicit bounds in Theorems 6.2–6.3 should be read as first quantitative estimates.
  • If the counting argument can be pushed to weaker differences, such as secants that barely exceed κ_φ on a thin set, then the range 1 < ab < π+1 for the classical Weierstrass function could become accessible, since the obstacle is only the verification of the (PD) condition in that range.
  • The slice statement dim_A W_y < 1 combined with the porosity of the graph suggests that level sets of these Weierstrass-type functions are strongly homogeneous; one could test computationally whether the bound in Theorem 1.4 is attained, which the paper does not address.
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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

0 major / 5 minor

Summary. The paper proves that for parameters a in (0,1) and b in N with ab >= pi+1, the Assouad dimension of the graph of the classical Weierstrass function W_{a,b} is strictly less than 2. The main tool is a generalized Weierstrass function W^phi_{a,b}; under a b-adic positive difference (PD) condition on the generator phi, the author proves dim_A G(W^phi_{a,b}) <= 2 - theta_phi < 2 (Theorem 4.2), with an explicit theta_phi, via the Aikawa dimension characterization. The cosine generator satisfies the PD condition for ab >= pi+1 and the sawtooth generator for ab >= 3, yielding Theorems 1.1 and 1.3; the latter answers negatively Yu's conjecture for Takagi functions in this range. The paper also gives quantitative bounds, porosity corollaries, and upper bounds for horizontal slices.

Significance. If correct, Theorem 1.1 resolves an open question of Fraser and overturns the prevailing heuristic, supported by typical Holder graphs and Wiener processes, that Weierstrass graphs should have Assouad dimension 2. The proof is notable for introducing a checkable b-adic PD condition and for deriving the bound with no fitted constants: theta_phi, I_phi, and L are constructed explicitly from the PD data. The counting argument in Section 3 and the Aikawa integral estimates in Section 4 are internally consistent, and the verifications for the cosine and sawtooth generators are explicit. The negative answer to Yu's conjecture for Takagi functions in the range ab >= 3 is a further concrete advance, as are the porosity and slice dimension corollaries.

minor comments (5)
  1. [Section 5.1, proof of Theorem 5.1] The display 'b^{n-p} < b r/R' is reversed; it should read 'b^{n-p} < b R/r'. The final bound N_r <= 10 b^{2-theta} C_0 (R/r)^{1-theta} follows from the corrected inequality, so this is a local typo rather than a substantive flaw.
  2. [Section 3, Lemma 3.1] The line 'Since P_n(J) is compact' is not justified for half-open intervals J in D_n, and the implication dist(y,P_n(J)) <= A a^n implies existence of x in J with |P_n(x)-y| <= A a^n can fail at boundary points. This is easily repaired by working with closures of the intervals or by a limiting argument with a slightly enlarged A and correspondingly adjusted M_A, but the step should be addressed in a revision.
  3. [Theorem 6.3] The proof of Theorem 6.3 is only a sketch ('Following similar arguments ... it can be shown'). Since the theorem states an explicit quantitative bound, please include the verification that n_phi=1, k_phi=floor(b/3), t_0=0 satisfy the PD condition, that I_phi=[1/150,1/6] satisfies (3.1), and that the chosen L satisfies (3.2).
  4. [Corollary 1.5] The range b in (a^{-1}(pi+1), infinity) should presumably be b >= a^{-1}(pi+1) to include the equality case ab = pi+1 covered by Theorem 1.1.
  5. [Various] There are several small presentation issues: 'we ahve' in the proof of Theorem 4.2, 'sp < theta theta' in Corollary 7.2, 'Cambridge Trats' in reference [21], and the phrase 'at most or equal to' in Theorem 6.2. The displayed formula in Theorem 6.2 is also difficult to parse; stating the bound as 2 - log(13/12)/(L log b) with L defined separately would improve readability.

Circularity Check

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No significant circularity; the dimension upper bound is derived from an explicit PD hypothesis that is verified directly for the cosine and sawtooth generators, with the external Aikawa/Assouad equivalence as the only imported theorem.

full rationale

The central claim dim_A G(W_{a,b})<2 is not reduced to its own input. Theorem 1.1 is obtained from Theorem 4.2, which is a conditional statement: if a Lipschitz 1-periodic phi satisfies the b-adic positive difference condition (PD), then dim_A G(W^phi_{a,b}) <= 2 - theta_phi. The PD condition is an explicit geometric hypothesis on truncated sums P_{n_phi}, not a restatement of the desired dimension bound, and it is verified for the cosine and sawtooth generators by the self-contained Proposition 6.1, whose proof uses only Lipschitz and oscillation estimates (equations (6.2)-(6.5)). The counting lemmas (Lemma 3.1, 3.2) derive cardinality bounds from PD without invoking the conclusion. Theorem 4.2 converts these bounds into an integral estimate for dist(z,G)^{-epsilon} via the Lehrbaeck-Tuominen Aikawa/Assouad equivalence, an external cited theorem [32], not a result of this paper. The explicit constants L, M_A, C in (3.2), (3.5), and Lemma 3.2 are chosen to satisfy the stated inequalities and depend only on a,b,phi,A; they are construction parameters, not fitted to the target bound. The self-citations [14] and [15] appear only as motivational or contextual examples and are not load-bearing in the proof chain. No parameter is fitted from data and renamed a prediction, and no uniqueness theorem from the author's prior work is invoked to force the argument. The paper is self-contained against the external benchmark of the Aikawa theorem and establishes its hypotheses for the concrete generators considered.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim does not introduce new physical or mathematical entities. It relies on two standard background theorems (Aikawa-Assouad equality, Lebesgue measure of graphs) and on the standing parameter range ab>1. The PD condition is a hypothesis of the theorem, not an invented entity.

assumptions (3)
  • standard math Aikawa dimension equals Assouad dimension for nonempty subsets of R^d (Theorem A, [32, Theorem 1.1]).
    Used in Theorem 4.2 to turn an integral estimate into a bound on dim_A.
  • standard math Lebesgue measure of the graph of a continuous function is zero.
    Invoked in Theorem 4.2 before applying monotone convergence to the distance integral.
  • domain assumption Standing range a in (0,1), integer b, ab>1, with phi Lipschitz and 1-periodic.
    Ensures W^phi_{a,b} converges, kappa_phi is well-defined, and the level L in Section 3 exists.

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Pith. "Pith review of On the Assouad dimension of Weierstrass function graphs." pith.science (2026). https://pith.science/paper/54KKBFAY

@misc{pith2026260811145,
  author       = {Pith},
  title        = {Pith review of: On the Assouad dimension of Weierstrass function graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/54KKBFAY}},
  note         = {Machine review of arXiv:2608.11145}
}
abstract

The class of Weierstrass functions $W_{a,b}$ is one of the first class of examples of continuous and nowhere differentiable real functions. A challenging line of research has been to determine the various dimensions of graphs $G(W_{a,b})$ of such functions. For instance, the Hausdorff dimension of $G(W_{a,b})$ was only recently determined by Shen in 2018, after a long series of partial results by many different authors. While the Assouad dimension of $G(W_{a,b})$ remains an open problem, also posed as a question by J. M. Fraser, there have been many indications that it might be equal to $2$. Such indications include the graph of Wiener processes, the graphs of almost all H\"older functions in the Baire category sense, and the graphs of Weierstrass functions after a series of countably many reflections all having Assouad dimension equal to $2$. In this paper we show that this is not the case, providing a quantitative upper bound on the Assouad dimension of $G(W_{a,b})$ that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions $W_{a,b}^\phi(x) = \sum_{j=0}^\infty a^j\phi(b^j x)$, which includes $W_{a,b}$ and the class of Takagi functions. The latter fact is used to also answer in the negative a conjecture of H. Yu on the Assouad dimension of graphs of Takagi functions for parameters $a\in (0,1)$, $b\in [3/a, \infty)\cap \mathbb{Z}$.

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Works this paper leans on

42 extracted references · 41 canonical work pages

  1. [1]

    Quasiadditivity of R iesz capacity

    Aikawa, H. Quasiadditivity of R iesz capacity. Math. Scand. 69 , 1 (1991), 15--30

    work page 1991

  2. [2]

    C., and Kawamura, K

    Allaart, P. C., and Kawamura, K. The T akagi function: a survey. Real Anal. Exchange 37 , 1 (2011/12), 1--54

    work page 2011

  3. [3]

    ack, J., Mudarra, C., and V\

    Anderson, T. C., Lehrb\"ack, J., Mudarra, C., and V\"ah\"akangas, A. V. Weakly porous sets and M uckenhoupt A_p distance functions. J. Funct. Anal. 287 , 8 (2024), Paper No. 110558, 34

    work page 2024

  4. [4]

    Slices of the T akagi function

    Anttila, R., B\'ar\'any, B., and K\"aenm\"aki, A. Slices of the T akagi function. Ergodic Theory Dynam. Systems 44 , 9 (2024), 2361--2398

    work page 2024

  5. [5]

    Espaces m\'etriques, plongements, facteurs , vol

    Assouad, P. Espaces m\'etriques, plongements, facteurs , vol. No. 223-7769 of Publications Math\'ematiques d'Orsay [Mathematical Publications of Orsay] . Universit\'e Paris XI, U.E.R. Math\'ematique, Orsay, 1977. Th\`ese de doctorat

    work page 1977

  6. [6]

    \'etude d'une dimension m\'etrique li\'ee \`a la possibilit\'e de plongements dans R n

    Assouad, P. \'etude d'une dimension m\'etrique li\'ee \`a la possibilit\'e de plongements dans R n . C. R. Acad. Sci. Paris S\'er. A-B 288 , 15 (1979), A731--A734

    work page 1979

  7. [7]

    Plongements lipschitziens dans R ^ n

    Assouad, P. Plongements lipschitziens dans R ^ n . Bull. Soc. Math. France 111 , 4 (1983), 429--448

    work page 1983

  8. [8]

    \"U ber die baire'sche kategorie gewisser funktionenmengen

    Banach, S. \"U ber die baire'sche kategorie gewisser funktionenmengen. Studia Mathematica 3\/ (1931), 174--179

    work page 1931

Show all 42 references
  1. [9]

    On the dimension of the graph of the classical W eierstrass function

    Bara\'nski, K., B\'ar\'any, B., and Romanowska, J. On the dimension of the graph of the classical W eierstrass function. Adv. Math. 265\/ (2014), 32--59

    2014

  2. [10]

    F., Harding, B., Vince, A., and Viswanathan, P

    Barnsley, M. F., Harding, B., Vince, A., and Viswanathan, P. Approximation of rough functions. Journal of Approximation Theory 209\/ (2016), 23--43

    2016

  3. [11]

    S., and Ursell, H

    Besicovitch, A. S., and Ursell, H. D. S ets of fractional dimensions ( V ): on dimensional numbers of some continuous curves. J. London Math. Soc. 1 , 1 (1937), 18--25

    1937

  4. [12]

    Continuous harmonic functions on a ball that are not in H^s for s>1/2

    Bramati, R., Dalla Riva, M., and Luczak, B. Continuous harmonic functions on a ball that are not in H^s for s>1/2 . The Journal of Geometric Analysis 36 , 6 (2026), 200

    2026

  5. [13]

    Hausdorff dimension of scale-sparse W eierstrass-type functions

    Carvalho, A. Hausdorff dimension of scale-sparse W eierstrass-type functions. Fund. Math. 213 , 1 (2011), 1--13

    2011

  6. [14]

    Chrontsios-Garitsis, E.-K., and Hildebrand, A. J. H\"older continuity and dimensions of fractal F ourier series. J. Anal. 33 , 4 (2025), 1973--1996

    2025

  7. [15]

    Chrontsios-Garitsis, E.-K., and Tyson, J. T. On the A ssouad spectrum of H \"older and S obolev graphs. Math. Proc. Cambridge Philos. Soc. 180 , 1 (2026), 105--131

    2026

  8. [16]

    J., Llorente, J

    Donaire, J. J., Llorente, J. G., and Nicolau, A. Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions. Revista Matem \'a tica Iberoamericana 30 , 3 (2014), 1037--1071

    2014

  9. [17]

    u rlichen functionen reeller argumente nach ihren aenderungen in den kleinsten intervallen. Journal f \

    du Bois-Reymond, P. Versuch einer classification der willk \"u rlichen functionen reeller argumente nach ihren aenderungen in den kleinsten intervallen. Journal f \"u r die reine und angewandte Mathematik 79\/ (1875), 21--37

  10. [18]

    a ck, J., Tuominen, H., and V \

    Dyda, B., Ihnatsyeva, L., Lehrb \"a ck, J., Tuominen, H., and V \"a h \"a kangas, A. V. Muckenhoupt A_p -properties of distance functions and applications to Hardy--Sobolev -type inequalities. Potential Anal. 50 , 1 (2019), 83--105

    2019

  11. [19]

    Fractal geometry , third ed

    Falconer, K. Fractal geometry , third ed. John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications

    2014

  12. [20]

    Feng, T., and Fraser, J. M. The A ssouad spectrum and dimension of typical graphs. Preprint: https://arxiv.org/abs/2601.05439

  13. [21]

    Assouad dimension and fractal geometry , vol

    Fraser, J. Assouad dimension and fractal geometry , vol. 222 of Cambridge Trats in Mathematics . Cambridge University Press, 2020

    2020

  14. [22]

    Hardy, G. H. Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17 , 3 (1916), 301--325

    1916

  15. [23]

    Lectures on analysis on metric spaces

    Heinonen, J. Lectures on analysis on metric spaces . Universitext. Springer-Verlag, New York, 2001

    2001

  16. [24]

    Weierstrass functions in Zygmund's class

    Heurteaux, Y. Weierstrass functions in Zygmund's class. Proceedings of the American Mathematical Society 133 , 9 (2005), 2711--2720

    2005

  17. [25]

    C., and Yu, H

    Howroyd, D. C., and Yu, H. Assouad dimension of random processes. Proc. Edinb. Math. Soc. (2) 62 , 1 (2019), 281--290

    2019

  18. [26]

    R., and Yorke, J

    Hunt, B. R., and Yorke, J. A. Smooth dynamics on Weierstrass nowhere differentiable curves. Transactions of the American Mathematical Society 325 , 1 (1991), 141--154

    1991

  19. [27]

    Ihnatsyeva, L., and V \"a h \"a kangas, A. V. Hardy inequalities in T riebel-- L izorkin spaces. II . A ikawa dimension. Ann. Mat. Pura Appl. (4) 194 , 2 (2015), 479--493

    2015

  20. [28]

    Assouad dimension of the T akagi function

    Jiang, L. Assouad dimension of the T akagi function. Preprint: https://arxiv.org/abs/2502.01140

    arXiv

  21. [29]

    L., Mallet-Paret, J., and Yorke, J

    Kaplan, J. L., Mallet-Paret, J., and Yorke, J. A. The L yapunov dimension of a nowhere differentiable attracting torus. Ergodic Theory Dynam. Systems 4 , 2 (1984), 261--281

    1984

  22. [30]

    Kuroda, S. , Ed. The collected papers of T eiji T akagi . Iwanami Shoten Publishers, Tokyo, 1973. With a preface by S. Iyanaga

    1973

  23. [31]

    Hardy inequalities and A ssouad dimensions

    Lehrb\"ack, J. Hardy inequalities and A ssouad dimensions. J. Anal. Math. 131\/ (2017), 367--398

    2017

  24. [32]

    A note on the dimensions of A ssouad and A ikawa

    Lehrb\"ack, J., and Tuominen, H. A note on the dimensions of A ssouad and A ikawa. J. Math. Soc. Japan 65 , 2 (2013), 343--356

    2013

  25. [33]

    Assouad dimension: antifractal metrization, porous sets, and homogeneous measures

    Luukkainen, J. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math.\ Soc. 35 , 1 (1998), 23--76

    1998

  26. [34]

    On the H ausdorff dimension of some fractal sets

    Przytycki, F., and Urba\'nski, M. On the H ausdorff dimension of some fractal sets. Studia Math. 93 , 2 (1989), 155--186

    1989

  27. [35]

    A dichotomy for the W eierstrass-type functions

    Ren, H., and Shen, W. A dichotomy for the W eierstrass-type functions. Invent. Math. 226 , 3 (2021), 1057--1100

    2021

  28. [36]

    Measure and Hausdorff dimension of randomized Weierstrass -type functions

    Romanowska, J. Measure and Hausdorff dimension of randomized Weierstrass -type functions. Nonlinearity 27 , 4 (2014), 787--801

    2014

  29. [37]

    On the p th variation of a class of fractal functions

    Schied, A., and Zhang, Z. On the p th variation of a class of fractal functions. Proceedings of the American Mathematical Society 148 , 12 (2020), 5399--5412

    2020

  30. [38]

    Hausdorff dimension of the graphs of the classical W eierstrass functions

    Shen, W. Hausdorff dimension of the graphs of the classical W eierstrass functions. Math. Z. 289 , 1-2 (2018), 223--266

    2018

  31. [39]

    Energy estimate for wave equations with coefficients in some Besov type class

    Tarama, S. Energy estimate for wave equations with coefficients in some Besov type class. Electronic Journal of Differential Equations 2007 , 85 (2007), 1--12

    2007

  32. [40]

    Volume estimates for unions of convex sets, and the K akeya set conjecture in three dimensions

    Wang, H., and Zahl, J. Volume estimates for unions of convex sets, and the K akeya set conjecture in three dimensions. Preprint: https://arxiv.org/abs/2502.17655

    arXiv

  33. [41]

    The A ssouad dimension of K akeya sets in R^3

    Wang, H., and Zahl, J. The A ssouad dimension of K akeya sets in R^3 . Invent. Math. 241 , 1 (2025), 153--206

    2025

  34. [42]

    Weak tangent and level sets of T akagi functions

    Yu, H. Weak tangent and level sets of T akagi functions. Monatsh. Math. 192 , 1 (2020), 249--264

    2020

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