Attainable forms of lower spectra

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 $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = \varphi(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*} \varphi(θ) \leq \varphi(λθ) - θ\varphi(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$. 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.

Figures (5)

• Figure 1: A depiction of the set constructed in the proof of \ref{['it:two-scale']}\ref{['i:ext']}. Each set $E_k$ is constructed as an intersection of nested families of dyadic cubes, following the procedure in \ref{['ss:unif-ct']}, with the subdivision sequence of each $E_k$ depending on $f_k$. The spacing between the consecutive sets is proportional to the diameter of $E_k$. The sets $E_k$ for $k \geq 3$ are omitted.
• Figure 2: The functions $\phi_{\alpha,\lambda_i, t_i}$
• Figure 3: The functions $\theta\mapsto \phi_{\alpha,\lambda_i, t_i}(\theta)/(1-\theta)$
• Figure 5: Plot of $q_{\bm{a}}$.
• Figure 6: Plot of $\theta\mapsto q_{\bm{a}}(\theta)/(1-\theta)$.

Theorems & Definitions (55)

• Theorem 1.1
• Definition 1.1
• Definition 1.2
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Lemma 2.2
• Proof 1
• Lemma 2.3