On the Assouad spectrum of Hölder and Sobolev graphs

Abstract We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\mathrm{Gr}({\kern2pt}f))$ 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.