Assouad type dimensions of homogeneous Moran sets and Cantor-like sets

TL;DR

This work derives an explicit Assouad dimension formula for homogeneous Moran sets under the condition $\sup_{k\ge1} n_k<\infty$, and provides an upper bound for the lower dimension. It also determines the Assouad spectrum and the lower spectrum for Cantor-like sets via a detailed use of the scale function $h(r)$ and the index $l(\theta,k)$, connecting growth of $\{n_k\}$ and contraction ratios $\{c_k\}$ to the spectra. The results generalize several prior findings on Moran and Cantor-like constructions, offering precise dimension characterizations for these fractal families and enabling sharper comparisons and embeddings in fractal geometry. The formulas furnish concrete tools for analyzing local density and sparsity patterns across scales, with potential implications for geometric measure theory and embedding problems in metric spaces.

Abstract

In this paper, we give the Assouad dimension formula and the upper bound of the lower dimension for homogeneous Moran sets under the condition $\sup_{k\ge 1}\{n_{k}\}<+\infty$. We also give the Assouad spectrum and the lower spectrum formulas for Cantor-like sets.

Theorems & Definitions (16)

• Definition 1
• Remark 1
• Remark 2
• Definition 2
• Remark 3
• Theorem 1
• Remark 4
• Theorem 2
• Theorem 3
• Theorem 4