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
TypeArticle
TitleGeneralised Cantor sets and the dimension of products
AuthorsOlson, 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.

LanguageEnglish
PublisherCambridge University Press
JournalMathematical Proceedings of the Cambridge Philosophical Society
ISSN0305-0041
Electronic1469-8064
Publication dates
Online30 Oct 2015
Print31 Jan 2016
Publication process dates
Deposited15 Oct 2015
Accepted29 Apr 2015
Output statusPublished
Accepted author manuscript
Copyright Statement

This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society
[http://doi.org/10.1017/S0305004115000584]. This version is free to view and download for private research and study only. Not for re-distribution or re-use. COPYRIGHT: © Cambridge Philosophical Society 2015

Digital Object Identifier (DOI)https://doi.org/10.1017/S0305004115000584
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