Generalised Cantor sets and the dimension of products

Olson, Eric J., Robinson, James C. and Sharples, Nicholas ORCID: https://orcid.org/0000-0003-1722-5647 (2016) Generalised Cantor sets and the dimension of products. Mathematical Proceedings of the Cambridge Philosophical Society, 160 (1) . pp. 51-75. ISSN 0305-0041 [Article] (doi:10.1017/S0305004115000584)

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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 cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. 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 α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ≥ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

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