The Assouad spectrum and dimension of typical graphs

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 $1$.