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Fraser's open problem on the dimension of the Takagi graph

For the classical Takagi function T:[0,1]→RT:[0,1]\to\mathbb{R}, T(x)=∑n=0∞2−ndist⁡(2nx,Z)T(x)=\sum_{n=0}^{\infty}2^{-n}\operatorname{dist}(2^n x,\mathbb{Z}), and the classical Weierstrass functions Wλ,N(x)=∑n=0∞λncos⁡(2πNnx)W_{\lambda,N}(x)=\sum_{n=0}^{\infty}\lambda^n\cos(2\pi N^n x), with N≥2N\ge 2 an integer and 0<λ<10<\lambda<1, determine the Assouad dimension dim⁡A\dim_A, quasi-Assouad dimension dim⁡qA\dim_{qA}, and Assouad spectrum dim⁡Aθ\dim_A^\theta, for θ∈(0,1)\theta\in(0,1), of their graphs ΓT\Gamma_T and ΓWλ,N\Gamma_{W_{\lambda,N}}.

References

Primary source

arXiv

Additional references

dimension-theoryfractal-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle the Takagi-graph dimension question, but the claim has not been independently checked.

Fraser’s open question asks for the Assouad dimensions, quasi-Assouad dimensions, and Assouad spectra of Takagi and Weierstrass graphs. The broader question remains only partially resolved in the retrieved literature.

Known results

  • For the classical base-rr Takagi function, dim⁡AGTr=1\dim_A\mathcal{G}_{T_r}=1 for every integer r≥2r\ge 2 (Anttila, Bárány, and Käenmäki, 2025).
  • For parametrized Takagi functions, dim⁡AGTa,b=1\dim_A G_{T_{a,b}}=1 exactly when 0≤a≤b−10\le a\le b^{-1} (Anttila, Bárány, and Käenmäki, 2025).
  • For b=2b=2 and a∈(1/2,1)a\in(1/2,1), a precise slice-dimension formula disproved Yu’s conjectured value 22 (Anttila, Bárány, and Käenmäki).
  • For a∈(0,1)a\in(0,1), b∈Nb\in\mathbb{N}, and ab≥3ab\ge 3, the Assouad dimension is strictly below 22 (Anttila, Bárány, and Käenmäki).

September 2026 claimed settlement

A preprint by Aaryan Dharmesh Shah and Sangita Jha claims exact Assouad-spectrum and quasi-Assouad-dimension results for the Takagi graph within a broader affine fractal-interpolation framework. If correct, this settles the Takagi portion of Fraser’s question, not the corresponding Weierstrass problem.

Current status (as of October 2026): The Takagi case is claimed solved by a September 2026 preprint, but that claim is unverified; the broader Weierstrass-related question remains open.

Sources

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