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 $\beta +\gamma \ge 1$ 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 ${\text{dim}}_{A}C={\text{dim}}_{A}D={\text{dim}}_A(C\times D)={\text{dim}}_B(C\times D)=\alpha$.

Cited by in corpus (4)

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• Prediction of dynamical systems from time-delayed measurements with self-intersections
• Regularity of non-autonomous self-similar sets