Attainable forms of Assouad spectra

Let $d\in \mathbb{N}$ and let $\phi \text{\hspace{0.17em}⁣}:(0,1)\to [0,d]$. We prove that there exists a set $F\subset {\mathbb{R}}^d$ such that ${\dim }_A^{\theta }F=\phi (\theta )$ for all $\theta \in (0,1)$ if and only if for every $0<\lambda <\theta <1$, $$0\le (1-\lambda )\phi (\lambda )-(1-\theta )\phi (\theta )\le (\theta -\lambda )\phi \left(\frac{\lambda }{\theta }\right)\mathrm{.}$$ 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.