On the Assouad spectrum of Hölder and Sobolev graphs

Abstract

We provide upper bounds for the Assouad spectrum ${\dim }_A^{\theta }(\text{Gr}(f))$ of the graph of a real-valued Hölder or Sobolev function $f$ defined on an interval $I\subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-Hölder continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta \in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

24 pages, 5 figures, To appear in Math. Proc. Cambridge Philos. Soc