On the Assouad spectrum of Hölder and Sobolev graphs

We provide upper bounds for the Assouad spectrum A^θ(Gr(2ptf)) of the graph of a real-valued Hölder 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 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∈ (0,1) . Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.