The Assouad Dimension of Graphs: An Affirmative Answer to Fraser's Question

Abstract

This article primarily investigates the Assouad dimension of graphs of continuous functions. Initially, the Assouad spectrum of the graph of a Hölder and count-Hölder continuous function has been derived. It has been shown that the Assouad dimension of the graph of a continuous function $f:[a,b]\to \mathbb{R}$ with the count-Hölder condition is at least \( 1 + \sup \left\{ \overline{\dim}_{\mathrm{B}}L_f(y) : y \in \mathscr{R}_f[a,b]\right\} \) where $\overline{\dim}_{\mathrm{B}}$ denotes the upper box dimension, $L_f(y)$ is the level set of $f$ at level $y$, and $\mathscr{R}_f[a,b]$ is the range of $f$ on $[a,b]$. As a consequence, non-trivial lower bounds are obtained for the Assouad dimension of the graphs of classical functions such as the Weierstrass and Takagi functions. We have also employed a unified projection solely in the direction of the $y$-axis to establish lower and upper bounds for the Assouad dimension. On this basis, we have shown that the lower bound is sharp in terms of level sets, namely the upper box dimension cannot be replaced by the Assouad dimension. This provides a partial positive answer to Question 17.11.1 posed by Fraser in~\cite{Fraser2021}, and to Question 2.2 posed by Yu in~\cite{Yu2020}. Building on this result, the analysis has been further extended to a broader class of Weierstrass-type functions and some self-affine functions like fractal interpolation functions. Moreover, we have discussed certain circumstances under which the Assouad dimension can reach 2 with several examples constructed. A supplementary explanation of why Brownian motion has the full Assouad dimension has also been provided based on the preceding work of Howroyd and Yu \cite{Howroyd2019}.