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2012/2013 Strict Inequality in the Box-Counting Dimension Product Formulas
James C. Robinson, Nicholas Sharples
Real Anal. Exchange 38(1): 95-120 (2012/2013).

Abstract

We supplement the well known upper and lower box-counting product inequalities to give the new product formula \begin{align*} \dim_{LB}F + \dim_{LB}G &\leq \dim_{LB}\left(F\times G\right) \\ &\leq \min\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ &\leq \max\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ & \leq \dim_{B}\left(F\times G\right)\\ &\leq \dim_{B}F+\dim_{B}G \end{align*} for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products.

Citation

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James C. Robinson. Nicholas Sharples. "Strict Inequality in the Box-Counting Dimension Product Formulas." Real Anal. Exchange 38 (1) 95 - 120, 2012/2013.

Information

Published: 2012/2013
First available in Project Euclid: 29 April 2013

zbMATH: 1277.28014
MathSciNet: MR3083200

Subjects:
Primary: 28A80

Keywords: box-counting dimension , Cartesian product , generalised Cantor set

Rights: Copyright © 2012 Michigan State University Press

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Vol.38 • No. 1 • 2012/2013
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