Generalised lower Assouad-type dimensions and their interpolations

TL;DR

This work defines and analyzes the φ-lower dimension $\dim_{\rm L}^{\phi}F$ and its quasi/modified variants in bounded doubling spaces, with the rate window mechanism $R^{1+\phi(R)}$ to capture fine-scale scaling. It proves a stability (equivalence) result when two dimension functions are asymptotically equivalent, develops a variational principle for rate windows, and establishes a positive interpolation result showing that any $s$ between $\dim_{\rm L}F$ and $\dim_{\rm qL}F$ can be realized by an appropriate $\phi$, while also presenting counterexamples where full interpolation to the lower box dimension fails. The paper further develops explicit Moran-set formulas to compute $\dim_{\rm L}^{\phi}$ in structured constructions and demonstrates, via popcorn-graph examples, that generalized lower dimensions do not always interpolate the entire scale from $\dim_{\rm L}$ to $\underline{\dim}_{\rm B}$. Overall, the results illuminate both the potential and limitations of φ-based interpolation in fractal dimension theory, offering a nuanced extension of lower Assouad-type dimensions.

Abstract

This paper investigates the analytic and structural properties of the $φ$-lower Assouad dimension, a generalized notion extending the lower Assouad dimension. We establish the equivalence of $φ$-lower Assouad dimensions with respect to the dimension functions, prove analytic properties related to the regularity of the $φ$-lower dimension, and analyse the role of rate windows in this context. Furthermore, we explore both positive and negative interpolation properties of the $φ$-lower dimension by presenting corresponding theorems that delineate these behaviors.

Figures (2)

• Figure 1: Figure of $R^{1+\phi(R)}$ for $R \in [R_{n+2}, R_n]$
• Figure 2: Figure of $R^{1+\phi(R)}$ for $R \in [R_{n+2}, R_n]$.

Theorems & Definitions (48)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Lemma 2: HT2019
• Lemma 3
• proof
• Lemma 4