Attainable forms of Assouad spectra

Abstract

Let $d\in \mathbb{N}$ and let $\ddot{I}†:(0,1)\to [0,d]$. We prove that there exists a set $F\subset {\mathbb{R}}^d$ such that ${\dim }_A^{\hat{I}}¸F=\ddot{I}†(\hat{I}¸)$ for all $\hat{I}¸\in (0,1)$ if and only if for every $0<\hat{I}»<\hat{I}¸<1$, \[0\leq (1-λ)φ(λ)-(1-θ)φ(θ)\leq (θ-λ)φ\Bigl(\fracλθ\Bigr).\] 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 Hölder in a neighbourhood of 1.

25 pages, 5 figures. v3: Many typo fixes and substantially improved exposition, especially in non-monotonic construction. Some numbering changes from v2. Results unchanged