Attainable forms of lower spectra

TL;DR

This paper characterizes which functions can be realized as the lower spectra of certain sets in Euclidean space, providing necessary and sufficient conditions and extending previous classifications to more general two-scale branching functions.

Contribution

It establishes a complete characterization of attainable lower spectra functions for sets in Euclidean space, extending prior results and employing a novel approach with two-scale branching functions.

Findings

Characterization of functions satisfying specific inequalities as attainable lower spectra.

Construction of sets with prescribed lower spectrum functions.

Extension of classification results beyond homogeneous sets.

Abstract

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