Generalised Cantor sets and the dimension of products

Olson, Eric J. and Robinson, James C. and Sharples, Nicholas (2016) Generalised Cantor sets and the dimension of products. Mathematical Proceedings of the Cambridge Philosophical Society, 160 (1). pp. 51-75. ISSN 0305-0041

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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.

Item Type: Article
Additional Information: Published online: 30 October 2015
Research Areas: A. > School of Science and Technology
Item ID: 18187
Useful Links:
Depositing User: Nicholas Sharples
Date Deposited: 15 Oct 2015 16:56
Last Modified: 06 Dec 2016 14:31
URI: http://eprints.mdx.ac.uk/id/eprint/18187

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