Generalised Cantor sets and the dimension of products
Article
Olson, E., Robinson, J. and Sharples, N. 2016. Generalised Cantor sets and the dimension of products. Mathematical Proceedings of the Cambridge Philosophical Society. 160 (1), pp. 51-75. https://doi.org/10.1017/S0305004115000584
Type | Article |
---|---|
Title | Generalised Cantor sets and the dimension of products |
Authors | Olson, E., Robinson, J. and Sharples, N. |
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. |
Publisher | Cambridge University Press |
Journal | Mathematical Proceedings of the Cambridge Philosophical Society |
ISSN | 0305-0041 |
Electronic | 1469-8064 |
Publication dates | |
Online | 30 Oct 2015 |
31 Jan 2016 | |
Publication process dates | |
Deposited | 15 Oct 2015 |
Accepted | 29 Apr 2015 |
Output status | Published |
Accepted author manuscript | |
Copyright Statement | This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society |
Digital Object Identifier (DOI) | https://doi.org/10.1017/S0305004115000584 |
Web of Science identifier | WOS:000366672600004 |
Language | English |
https://repository.mdx.ac.uk/item/85zv5
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