Generalised Cantor sets and the dimension of products

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 $\text{dim}_{A}C=\text{dim}_{A}D=\text{dim}_{A}(C\times D)=\text{dim}_{B}(C\times D)=\alpha$.