On the Assouad dimension of Weierstrass function graphs

[2608.11145] On the Assouad dimension of Weierstrass function graphs Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2608.11145 (math) [Submitted on 11 Aug 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: On the Assouad dimension of Weierstrass function graphs Authors: Efstathios Konstantinos Chrontsios Garitsis View PDF HTML (experimental) 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 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 $W_{a,b}^\phi(x) = \sum_{j=0}^\infty a^j\phi(b^j x)$, 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 $a\in (0,1)$, $b\in (1/a, \infty)\cap \mathbb{Z}$. Comments: 26 pages, 1 figure Subjects: Dynamical Systems (math.DS) ; Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG) Cite as: arXiv:2608.11145 [math.DS] (or arXiv:2608.11145v2 [math.DS] for this version) https://doi.org/10.48550/arXiv.2608.11145 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Efstathios Konstantinos Chrontsios Garitsis [ view email ] [v1] Tue, 11 Aug 2026 17:04:13 UTC (24 KB) [v2] Fri, 28 Aug 2026 09:32:50 UTC (325 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-08 Change to browse by: math math.CA math.MG References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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