The Assouad spectrum and dimension of typical graphs
Tianyi Feng, Jonathan Fraser
TL;DR
Using Baire category in the space $c^\omega[0,1]$, the authors determine the typical Assouad dimension and Assouad spectrum of graphs $G_f = \{(x,f(x))\}$ for functions in various Banach spaces. They derive a universal bound: for a modulus of continuity $\omega$ with $\eta = \liminf_{t\to0} \frac{\log \omega(t)}{\log t}$, and $f\in C^\omega[0,1]$, $\dim_A^{\theta} G_f \le 2 - \frac{\eta-\theta}{1-\theta}$ for $\theta \in (0,\eta)$, with sharpness in the concave case; moreover, if $\eta \ge 1$ then $\dim_A^{\theta} G_f = 1$ for all $\theta$. In the little spaces $c^\omega[0,1]$ with concave $\omega$ and $\eta<1$, a typical $f$ attains $\dim_A^{\theta} G_f = 2 - \frac{\eta-\theta}{1-\theta}$ for $\theta \le \eta$ and $2$ for $\theta > \eta$, while the quasi-Assouad dimension is $1$ when $\eta \ge 1$ and $2$ when $\eta<1$. In particular, for little $\alpha$-Hölder spaces ($\omega(t)=t^{\alpha}$) typical graphs have $\dim_A G_f = 2$ and $\dim_A^{\theta} G_f = 2 - \frac{\alpha-\theta}{1-\theta}$ for $\theta \le \alpha$, with saturation beyond, and for the log-modulated case $\omega(t)=t(1+|\log t|)$ the quasi-Assouad spectrum collapses to $1$ while the full Assouad dimension remains $2$.
Abstract
We investigate the Assouad spectrum and dimension of graphs of functions lying in certain Banach spaces. We find the typical values in the sense of Baire category, proving that these values are often as large as possible, given the constraints of the particular function space. For example, we demonstrate that in the little $α$-Hölder spaces, a typical graph has Assouad dimension $2$ and Assouad spectrum $\min\{2,2-\frac{α-θ}{1-θ}\}$; whereas in the space associated with modulus of continuity $t(1+|\log t|)$, a typical graph has Assouad dimension $2$ but quasi-Assouad dimension (and Assouad spectrum) equal to $1$.