Attainable forms of Assouad spectra

TL;DR

This paper characterizes which functions can be realized as the Assouad spectrum of a set in Euclidean space, revealing new possible behaviors such as non-monotonicity and failure of Hölder continuity near 1.

Contribution

It provides necessary and sufficient conditions for a function to be the Assouad spectrum of some set, including novel behaviors not previously observed.

Findings

Assouad spectrum can be non-monotonic on open sets

Assouad spectrum can fail to be Hölder near 1

Characterization of attainable Assouad spectra functions

Abstract

Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^\theta F=\varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for every $0<\lambda<\theta<1$, \[0\leq (1-\lambda)\varphi(\lambda)-(1-\theta)\varphi(\theta)\leq (\theta-\lambda)\varphi\Bigl(\frac{\lambda}{\theta}\Bigr).\] In particular, the following behaviours which have not previously been witnessed in any examples are possible: the Assouad spectrum can be non-monotonic on every open set, and can fail to be H\"older in a neighbourhood of 1.