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

TL;DR

This paper derives formulas for Assouad and spectrum dimensions of homogeneous Moran and Cantor-like sets, providing new insights into their fractal geometric properties under specific conditions.

Contribution

It introduces explicit formulas for Assouad and spectrum dimensions of these sets, expanding understanding of their fractal dimensions in the context of Moran and Cantor-like constructions.

Findings

Assouad dimension formula for homogeneous Moran sets

Upper bound of lower dimension for these sets

Formulas for Assouad and lower spectra of Cantor-like sets

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.