Generalised Cantor sets and the dimension of products

TL;DR

This paper explores the relationship between Assouad and box-counting dimensions, introduces equi-homogeneity, and constructs generalized Cantor sets demonstrating specific dimension behaviors under products.

Contribution

It establishes conditions under which Assouad and box-counting dimensions coincide and constructs examples of Cantor sets with prescribed product dimension properties.

Findings

Assouad and box-counting dimensions coincide for equi-homogeneous sets with equal upper and lower box-counting dimensions.

Constructed Cantor sets with specified box-counting dimensions and equal Assouad dimensions.

Demonstrated that product dimensions can be controlled independently of individual set dimensions.

Abstract

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of `equi-homogeneity' of a set, which requires a uniformity in the size of local covers at all lengths and at all points. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set `attains' these dimensions (analogous to `s-sets' when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any $\alpha\in(0,1)$ and any $\beta,\gamma\in(0,1)$ such that $\beta + \gamma\geq 1$ we can construct two generalised Cantor sets $C$ and $D$ such that $\text{dim}_{B}C=\alpha\beta$, $\text{dim}_{B}D=\alpha\gamma$, and $\text{dim}_{A}C=\text{dim}_{A}D=\text{dim}_{A}(C\times D)=\text{dim}_{B}(C\times D)=\alpha$.