On the Assouad spectrum of Hölder and Sobolev graphs

We provide upper bounds for the Assouad spectrum $\dim_A^θ(\text{Gr}(f))$ 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 $α$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $α$ and returns the graph of a new $α$-Hölder continuous function for which the Assouad $θ$-spectrum realizes the stated upper bound for all $θ\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.