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Attainable forms of Assouad spectra

Alex Rutar

TL;DR

This work completely characterizes which θ→φ(θ) can be realized as the Assouad spectrum dim_A^θ F of a subset F⊂ℝ^d. The authors introduce the admissible function class 𝒜_d and prove that φ sits in 𝒜_d exactly when 0 ≤ (1−λ)φ(λ) − (1−θ)φ(θ) ≤ (θ−λ)φ(λ/θ) for all 0<λ<θ<1, establishing rate and oscillation constraints that govern realizability. They develop a Moran-set construction framework to realize any admissible φ and analyze two main subfamilies (monotonic and non-monotonic spectra), plus closure properties and a complete description of upper Assouad spectra. The paper further demonstrates that exceptional spectra can exhibit Hölder failure at 1 and non-monotonicity on every open set, and shows these phenomena are dense in the spectrum-annotated landscape. Overall, the results provide a comprehensive, constructive classification of attainable Assouad spectra with implications for intermediate dimensions and related scaling theories.

Abstract

Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that there exists a set $F\subset\mathbb{R}^d$ such that $\operatorname{dim}_A^θF=\varphi(θ)$ for all $θ\in(0,1)$ if and only if for every $0<λ<θ<1$, \[0\leq (1-λ)\varphi(λ)-(1-θ)\varphi(θ)\leq (θ-λ)\varphi\Bigl(\fracλθ\Bigr).\] 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.

Attainable forms of Assouad spectra

TL;DR

This work completely characterizes which θ→φ(θ) can be realized as the Assouad spectrum dim_A^θ F of a subset F⊂ℝ^d. The authors introduce the admissible function class 𝒜_d and prove that φ sits in 𝒜_d exactly when 0 ≤ (1−λ)φ(λ) − (1−θ)φ(θ) ≤ (θ−λ)φ(λ/θ) for all 0<λ<θ<1, establishing rate and oscillation constraints that govern realizability. They develop a Moran-set construction framework to realize any admissible φ and analyze two main subfamilies (monotonic and non-monotonic spectra), plus closure properties and a complete description of upper Assouad spectra. The paper further demonstrates that exceptional spectra can exhibit Hölder failure at 1 and non-monotonicity on every open set, and shows these phenomena are dense in the spectrum-annotated landscape. Overall, the results provide a comprehensive, constructive classification of attainable Assouad spectra with implications for intermediate dimensions and related scaling theories.

Abstract

Let and let . We prove that there exists a set such that for all if and only if for every , 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.
Paper Structure (19 sections, 21 theorems, 83 equations, 5 figures)

This paper contains 19 sections, 21 theorems, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Classifying Assouad spectra
  3. Rate constraints and the relationship with intermediate dimensions
  4. Structure and outline of the paper
  5. Forms of the family of functions Ad
  6. Rescaling and a geometric interpretation of the bound
  7. Basic properties
  8. Rate constraints and increasing functions
  9. Classifying the forms of Assouad spectra
  10. Bounding the Assouad spectrum
  11. Constructing sets with prescribed spectra
  12. Families of monotonic and non-monotonic spectra
  13. Monotonic spectra
  14. Non-monotonic spectra
  15. Closure under suprema
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $d\in\mathop{\mathrm{\mathbb{N}}}\nolimits$ and let $\varphi\colon(0,1)\to[0,d]$ be a function. Then there exists $F\subset\mathop{\mathrm{\mathbb{R}}}\nolimits^d$ such that $\operatorname{dim}^{\theta}_{\mathrm{A}} F=\varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for every $0<\lambda<

Figures (5)

  • Figure 1: A plot of $\beta(\theta)=(1-\theta)\varphi(\theta)$ where $\varphi\in\mathcal{A}_d$, and the lines with slopes corresponding to \ref{['e:h-bd']}.
  • Figure 2: The concatenation of $(f_1,e_1,f_2)$ corresponding to a function $\phi\in\mathcal{C}_d$ defined in \ref{['ss:non-mono-ex']} restricted to the domain $(0,\infty)$.
  • Figure 3: A plot of $f_{\bm{i}}(\theta)/(1-\theta)$ and $h_{\bm{c}}(\theta)/(1-\theta)$ where $\bm{i}=(\kappa,c_2)$ and $\bm{c}=(\kappa,c_1,c_2)$.
  • Figure 4: Plot of a spectrum which is not Hölder at $1$, along with the general upper bound.
  • Figure 5: A plot of a zigzagging family for $\varphi(\theta)=1+\theta$, with $\mathcal{L}=\{1/4,1/2\}$, $t=2$, and an appropriate choice of $y$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.1: zbl:1509.28005
  • Definition 2.1
  • Lemma 2.2
  • Proof 1
  • Proposition 2.3
  • ...and 33 more