Attainable forms of lower spectra

Abstract

Let $d\in \mathbb{N}$ and $\ddot{I}†:(0,1)\to [0,d]$. We prove there exists a set $F\subset {\mathbb{R}}^d$ whose lower spectrum ${\dim }^{\hat{I}}{¸}_{\mathrm{L}}F$ satisfies $(1-\hat{I}¸){\dim }^{\hat{I}}{¸}_{\mathrm{L}}F=\ddot{I}†(\hat{I}¸)$ for all $\hat{I}¸\in (0,1)$ if and only if for all $\hat{I}»,\hat{I}¸\in (0,1)$, \begin{equation*} φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for ${\underline{\dim }}^{\hat{I}}{¸}_{\mathrm{L}}F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.

30 pages, 3 figures