On the Assouad dimension of Weierstrass function graphs
Abstract
The class of Weierstrass functions Wa,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(Wa,b) of such functions. For instance, the Hausdorff dimension of G(Wa,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(Wa,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 Hölder 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(Wa,b) that is strictly less than $2$. In particular, we show that such a bound is true for a class of generalized Weierstrass functions Wa,bϕ(x) = ∑j=0∞ ajϕ(bj x), which includes Wa,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∈ (0,1), b∈ (1/a, ∞)∩ ℤ.
Related
- Assouad Dimensions and Spectra of Graphs of Weierstrass-Type Functions with Rapidly Growing Frequencies
- The Assouad Dimension of Graphs: An Affirmative Answer to Fraser's Question
- Holomorphic motions, Assouad dimension and quasiconformal mappings
- Assouad--Nagata dimension of systolic complexes with almost flat intervals
- Orthogonal Projections and Assouad Dimension: Resolving Fraser's Questions
BibTeX
@misc{garitsis2026assouad,
title = {{On the Assouad dimension of Weierstrass function graphs}},
author = {Efstathios Konstantinos Chrontsios Garitsis},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.11145},
eprint = {2608.11145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11145},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Assouad dimension of Weierstrass function graphs
AU - Efstathios Konstantinos Chrontsios Garitsis
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.11145
UR - https://arxiv.org/abs/2608.11145
AB - 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 Hölder 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^ϕ(x) = ∑_j=0^∞ a^jϕ(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∈ (0,1), b∈ (1/a, ∞)∩ ℤ.
ER -