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Generalised Cantor sets and the dimension of products

Published online by Cambridge University Press:  30 October 2015

ERIC J. OLSON ,
JAMES C. ROBINSON  and
NICHOLAS SHARPLES
ERIC J. OLSON
Affiliation:
Mathematics and Statistics, University of Nevada, Reno, NV, 89507, U.S.A. e-mail: ejolson@unr.edu
JAMES C. ROBINSON
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV47AL. e-mail: j.c.robinson@warwick.ac.uk
NICHOLAS SHARPLES
Affiliation:
Department of Mathematics, Imperial College London, London, UK, SW72AZ e-mail: nicholas.sharples@gmail.com
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 51 - 75
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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