On the Assouad spectrum of H\"older and Sobolev graphs

TL;DR

This paper establishes sharp upper bounds for the Assouad spectrum of graphs of H"older and Sobolev functions, and introduces a geometric algorithm to realize these bounds for certain H"older functions, including classical fractal examples.

Contribution

It provides the first sharp bounds for the Assouad spectrum of H"older and Sobolev graphs and introduces a constructive algorithm for H"older functions satisfying specific oscillation conditions.

Findings

Upper bounds for the Assouad spectrum are sharp and optimal.

The geometric algorithm constructs functions achieving the bounds.

Classical fractal functions like Weierstrass and Takagi are included.

Abstract

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\text{Gr}(f))$ of the graph of a real-valued H\"older 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 H\"older graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-H\"older continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-H\"older 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.