@article{aea3128a4795488b9ed38726b4650f37,
title = "On the Assouad spectrum of H{\"o}lder and Sobolev graphs",
abstract = "We provide upper bounds for the Assouad spectrum dimθA (Gr( f)) of the graph of a real-valued H{\"o}lder or Sobolev function f defined on an interval I ⊂ R. We demonstrate via examples that all of our bounds are sharp. In the setting of H{\"o}lder graphs, we further provide a geometric algorithm which takes as input the graph of an α -H{\"o}lder continuous function satisfying a matching lower oscillation condition with exponent α and returns the graph of a new α -H{\"o}lder continuous function for which the Assouad θ -spectrum realizes the stated upper bound for all θ ∈ (0, 1). Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.",
author = "Chrontsios-Garitsis, \{Efstathios K.\} and Tyson, \{Jeremy T.\}",
note = "We gratefully acknowledge valuable insights from Jonathan Fraser on the subject of Assouad dimensions of graphs, and especially for bringing to our attention the recent preprint [] on the Assouad dimensions of Takagi graphs. We also thank Roope Anttila for pointing out that an earlier, weaker version of Question was in fact known. JTT acknowledges support from the Simons Foundation under grant \#852888, {\textquoteleft}Geometric mapping theory and geometric measure theory in sub-Riemannian and metric spaces{\textquoteright}. In addition, this material is based upon work supported by, and while JTT was serving as a Program Director at, the U.S. National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Lastly, we wish to thank the anonymous referee for reading our manuscript carefully, and for providing insightful feedback that has improved the exposition.",
year = "2026",
month = jan,
day = "1",
doi = "10.1017/S0305004125101527",
language = "English (US)",
volume = "180",
pages = "105--131",
journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
issn = "0305-0041",
publisher = "Cambridge University Press",
number = "1",
}