The Assouad spectrum and dimension of typical graphs

TL;DR

This paper studies the Assouad spectrum and dimension of typical graphs of functions in specific Banach spaces, revealing that most graphs have maximal or near-maximal dimensions under certain conditions.

Contribution

It characterizes the typical Assouad spectrum and dimension of graphs in various Banach spaces, showing they often reach the upper bounds dictated by the space constraints.

Findings

Typical graphs in little α-Hölder spaces have Assouad dimension 2.

In certain modulus of continuity spaces, typical graphs have Assouad dimension 2 but lower quasi-Assouad dimension.

Assouad spectrum varies with the function space and parameters, often reaching maximum values.

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 $\alpha$-H\"older spaces, a typical graph has Assouad dimension $2$ and Assouad spectrum $\min\{2,2-\frac{\alpha-\theta}{1-\theta}\}$; 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 $1$.